I play a lot of the Pokemon Trainer. The versatility is awesome. Here's my comments on matchup balance for each of the three Pokemon.
Squirtle (aka Ninja Turtle) Vs
Bowser, advantage: neutral. Bowser can kill Squirtle at ridiculously low damage, but he might have a hard time hitting him. Unlike most characters, Squirtle doesn't get any particular advantage against large targets.
Captain Falcon, advantage: ??? Who plays Falcon? No one I know.
Charizard, advantage: Charizard. Rock smash is useless here. Squirtle can DI out of fire breath quickly. That's about all Squirtle has on Charizard. Squirtle has to deal a LOT of damage to Charizard to budge him, while all Charizard has to do is land a couple attacks in and Squirtle goes flying.
Diddy Kong, advantage: ??? I play as both characters, but haven't played them against each other enough to comment.
Donkey Kong, advantage: ??? DK has lousy aerials but huge reach on his attacks. Not sure about this one.
Falco, advantage: Falco. The shine will bring Squirtle's SH f-air game to a screeching halt, so he can't close distance effectively. He's going to have to, otherwise Squirtle will be eating lasers.
Fox, advantage: ??? I haven't done this match much. Fox's u-smash could possibly obliterate Squirtle.
Mr. G&W, advantage: neutral. Assuming that G&W uses his b-air (which is a turtle, ironically enough), this could be a very even fight.
Ganondorf, advantage: Ganondorf. No joke. Although Ganon is slow, his attacks have huge power and have the capability of destroying Squirtle. Squirtle's best bet is to swap out for Ivysaur.
Ice Climbers, advantage: ICs. Squirtle can't be chain grabbed, but those hammers will slaughter Squirtle. All of ICs offensive specials really mess Squirtle up.
Ike, advantage: Ike. All of Ike's attacks have KO potential. Not to mention, his attacks have range, priority, a disjointed hitbox, and damage. The only thing Squirtle can do is keep out of range and jump in if an opportunity presents itself (or just swap out).
Ivysaur, advantage: Ivysaur. Razor Leaf forces Squirtle to keep mobile. Bullet Seed is over powered. Ivysaur's u-smash will absolutely annihilate a Squirtle.
Jigglypuff, advantage: Jigglypuff. Jigglypuff has better air Control and can easily spam f-air. Both characters have roughly the same battle strategy, though Jigglypuff does it better. The only thing Squirtle really has on a Jigglypuff is a killer u-smash (which most Jpuffs can easily avoid anyway).
King Dedede, advantage: Squirtle. Dedede is big and bad, but has troubles with ninja turtles like Squirtle. Dedede's famous chain grab is useless here. Thrown minions will have trouble hitting a target with such a tiny hit box. If Dedede ever does manage to get a f-smash in, he'll turn Squirtle in to turtle soup (which doesn't happen often).
Kirby, advantage: Kirby. Final Cutter can give Squirtle a rough time. Kirby fights much better on the ground than Squirtle does.
Link, advantage: Squirtle. Link's projectile game isn't very effective here. Squirtle is very fast and has an in-your-face play style that forces Link to use silly attacks like the up-b spin and clawshot spam.
Lucario, advantage: Squirtle. Things look very bad for Lucario. It's hard to hit Squirtle with the charge orb, counter attack usually fails because of Squirtle's speed, d-air is ridiculously hard to land, and Lucario's other aerial attacks aren't powerful enough to take on Squirtle's agility.
Lucas, advantage: Squirtle. The Squirtle will have to play pro to beat a Lucas, but he is fast enough to do so. Squirtle is fast enough to continuously deal aerial attacks to knock Lucas out and evade spells. Lucas has a huge delay after failing to land u-smash or a throw that Squirtle can jump all over and punish.
Luigi, advantage: ??? No idea.
Mario, advantage: ??? Again, no idea. Water Gun is still way cooler than FLUDD.
Marth, advantage: Marth. Marth has insane arial attacks (n-air f-air ftw). Marth has long range, disjointed hitbox, better priority, and a f-smash that ends games. Bye Squirtle.
MetaKnight, advantage: MetaKnight. Any of MK's specials (except that useless Vanish move) cause Squirtles massive problems. Also, MetaKnight is one of the dreaded sword wielding characters in the game. Swords = priority = turtle soup. Swap to Ivysaur and destroy MK with a u-smash.
Ness, advantage: Squirtle. See the Lucas description, except there's a twist here; Ness's u-smash yoyo has been buffed since Melee. This attack is annoying to get around. Ness is generally slow and very floaty in the air, making him a prime target for Squirtle.
Peach, advantage: ??? No one seems to play Peach any more.
Pikachu, advantage: Pikachu. Pikachu's f-smash is ridiculously effective against Squirtle. Pikachu is fast enough to evade Squirtle and punish him with constant lightning attacks.
Pikmin & Olimar, advantage: Squirtle. Olimar's standard strategy is to make use of his insane grab, then make use of f-smash and u-smash. If you attack Olimar at an angle, he's in trouble. Sure, he can up-B to shoot up, but if he misses (or if Squirtle dodges), he is open to attack for a long time. Many Olimar's don't use this ability because of it's unreliability (you must have 6 Pikmin standing right next to you). Squirtle attacks in just such an angle that Olimar's play style is ineffective.
Pit, advantage: Pit. Another dreaded sword user. To make matters worse, Pit is fast, has an insane projectile, and possibly the best recovery in the game. He's such a ridiculously over powered character. Gah. Pit's dashing u-smash will end Squirtle.
R.O.B., advantage: neutral. ROB might have a hard time landing lasers and gyros, but is able to annihilate Squirtle with any of his smash attacks.
Samus, advantage: Squirtle. Samus's fat missiles will have a hard time hitting tiny targets, so Samus will have to use standard missiles. Samus might not be able to make uses of missiles at all if the Squirtle is all up in her business. Sure, the charge shot will destroy Squirtle, but she'll likely never get a chance to charge it with the ninja turtle bouncing around.
Shiek, advantage: neutral. Shiek's f-tilt combo doesn't work very well against Squirtle. Needles do.
Snake, advantage: Snake. Snake's melee combos all have great knockback. Once the Squirtle's been knocked back, Snake can modify the battle stage with mines and traps. Snake's explosives have crazy knockback (bad news for lightweights) and his b-air will end any Squirtle it hits.
Sonic, advantage: neutral. Sonic can deal with fast characters.
Toon Link, advantage: ??? Not sure about this one; TL plays differently then Link. TL's insane d-air and boomerang might mess Squirtle up.
Wario, advantage: Wario. Wario's air control is insane. Squirtle has a faster game when playing on the land, but Wario's attacks will ruin our blue turtle.
Wolf, advantage: Wolf. Wolf's projectile (it has such a ridiculously huge hit box) will give Squirtle a rough time, all while replenishing Wolf's very powerful A attacks.
Yoshi, advantage: ??? When it comes to Yoshi, I never know.
Zelda, advantage: Zelda. Zelda's u-smash will actually catch an aerial Squirtle, suck him in to a vortex of fairy sparkle death, then shoot him to the top of the stage for an easy kill. Zelda's dash attack also wrecks Squirtle. Din's Fire forces Squirtle to keep aggressive.
Zero Suit Samus, advantage: neutral. ZSS will have a hard time trying to land the forward-b kill move. ZSS's f-smash has crazy AoE, hitting things behind you, so Squirtle will have to get his ninja dodging on.
Ivysaur (aka Death From Below) Vs
Bowser, advantage: Ivysaur. Bullet Seed is so powerful!! Run up to Bowser, hold B, Bowser takes ~80% damage. What the hell. Since Bowser is such a fat target, he will have a nightmare trying to DI out of Bullet Seed.
Captain Falcon, advantage: ??? Not enough Falcon players.
Charizard, advantage: Charizard. Flamethrower is very easy to use against Ivysaur. Since Ivysaur is such a lightweight, Charizard can easily land a KO after cooking Ivysaur. Ivysaur's only chance is to go for a u-smash KO.
Diddy Kong, advantage: Diddy. Bananas are justice. Jump and throw a banana forward, use a dash attack to hit Ivysaur and collect the banana again, throw it at Ivy again. Ivysaur will have to spam Razor Leaf to deal any damage.
Donkey Kong, advantage: Ivysaur. Bullet Seed, GO! DK can't do anything about Razor Leaf besides evade.
Falco, advantage: Falco. Falco wins here by taking advantage of Ivysaur's horrible recovery. Edgehog ftw.
Fox, advantage: Fox. The hardest thing for Fox here is the approach. Once he's able to attack Ivysaur, it's over (if Ivy has some damage on him). The laser can deliver damage.
Mr. G&W, advantage: ??? Can the bucket catch Razor Leafs? I'm not sure how this one will turn out.
Ganondorf, advantage: neutral. Razor Leaf spam, and if you're lucky enough to get under him, Bullet Seed. Ivysaur will probably end up using a lot of techroll grab attacks on Ganon. Ganon's sweep kick d-smash will hurt Ivy a lot, but he's too slow to use it often.
Ice Climbers, advantage: neutral. Ivysaur's f-tilt will work great for slicing these two. Bullet Seed isn't as effective, because if one of the two get free, they get a free hammer attack on Ivy. Razor Leaf is useful, but the usefulness of it is negated by IC's ice block projectile.
Ike, advantage: neutral. Ivysaur is quick enough to avoid Ike's attacks while attacking. Most of the time.
Jigglypuff, advantage: Ivysaur. Jigglypuff is a floaty character. Floaty characters die easily by being shot straight up. Ivysaur's u-smash is the strongest in the game. Jigglypuff is the lightest weight character. Game.
King Dedede, advantage: King Dedede. Ivysaur has a surprisingly hard time approaching Dedede. Obviously Bullet Seed will be your prime advantage, but getting close enough to use it is another matter. Razor Leaf bounces uselessly through the minions, and if you try to get close, Ivy ends up eating a hammer.
Kirby, advantage: ??? Not sure about this one.
Link, advantage: ??? I'd rather use Squirtle.
Lucario, advantage: ??? Bullet Seed beats Lucario's d-air, but what about the land combat?
Lucas, advantage: Lucas. PK Fire equals razor leaf in terms of priority, but PK Fire casts much faster. Ivysaur will have to resort to other attacks. Lucas can easily escape bullet seed. Ivysaur has a horrible recovery and is easily edgeguarded.
Luigi, advantage: ??? The Luigi Missile might be an interesting approach, but I don't know how it'd interact with Ivy's attacks.
Mario, advantage: ??? I don't play against Mario often.
Marth, advantage: Marth. Marth will have to close distance quickly due to Razor Leaf. Once in close distance, Marth has a number of advantages over Ivysaur. Dragon Slayer (f-smash) will annihilate Ivy, and Marth can quickly move to edgehog, foiling Ivy's tether recovery.
MetaKnight, advantage: Ivysaur. A floaty character! Use Razor Leaf to force him to close distance, use throws and Bullet Seed to rack up damage, then use u-smash to get a sure kill. Horizontal smashes are useless against MetaKnight unless the hit stun lasts long enough for him to be wiped out.
Ness, advantage: Ness. PK Fire absolutely owns Ivysaur. Razor Leaf isn't going to help you here. Ness can combo PK Fire to baseball bat f-smash easy, devastating Ivysaur.
Peach, advantage: ??? I don't play against competent Peaches much.
Pikachu, advantage: neutral. Pikachu will probably refrain from using arial attacks unless Ivy jumps first. Projectile spamathon.
Pikmin & Olimar, advantage: neutral. Razor Leaf doesn't cut through Pikmin, but Pikmin are a limited projectile. Olimar is more likely to run up to Ivysaur and toss him around with his superior throw. Olimars are never approaching by air, so Ivy's u-smash isn't likely going to get much use.
Pit, advantage: Pit. Pit is fast, has an insane projectile, and nearly infinite recovery. Not to mention, he has two skills that deflect projectiles. Pit's f-smash comes out so fast, you can't react to it at all. You also can't punish Pit when he recovers from a missed smash since he has barely any lag. Blah.
R.O.B., advantage: neutral. Razor Leaf can stop gyros, but that doesn't mean Ivy's safe from lasers. ROB is very susceptible to Bullet Seed, though watch out for ROB's d-smash; it's like Shiek's d-smash from melee (fast, easy to hit, damaging, knockbacktastic, minimal lag).
Samus, advantage: Samus. Homing missiles can be shot from higher ground so Razor Leaf can't intercept them. If you're under Samus, she'll probably drop some bombs for you to play with. The charge cannon will destroy Ivy at low percentages. Samus also has a better throw and aerial combat ability.
Shiek, advantage: neutral. Shiek's f-tilt combo works marginally well against Ivy. Use Razor Leafs to force Shiek in the air, then try to get some hits in with either Bullet Seed or u-smash.
Snake, advantage: neutral. Ivy might have a hard time dealing damage. Snake will likely keep trying to run away, throwing nades and planting explosives. Remember that Ivy's up-b is also a great attack.
Sonic, advantage: Ivysaur. Sonic's rolls have such horrible priority. Ivy's have an easy time spamming projectiles and using grabs.
Squirtle, advantage: Ivysaur. Razor Leaf forces Squirtle to keep mobile. Ivysaur's u-smash will absolutely annihilate a Squirtle. Hell, even Ivysaur's f-smash could wreck a Squirtle.
Toon Link, advantage: ??? Not sure.
Wario, advantage: ??? Probably in Wario's favor. Wario's air game could throw Ivy way off. Wario can easily be grabbed while on the hog.
Wolf, advantage: Wolf. Wolf's strong ground-based attacks will hurt Ivy. Plus, Wolf has a reflector.
Yoshi, advantage: ??? Yoshi turns in to an egg and rolls off the stage.
Zelda, advantage: neutral. Zelda's attacks all have range, priority, and disjointed hitboxes. The idea here is to play mind games with Razor Leaf so Zelda keeps miscasting the deflection spell.
Zero Suit Samus, advantage: ZSS. This is a horrible matchup for Ivysaur. ZSS's forward-b will murder Ivy.
Charizard (aka Stock Tank) Vs
Bowser, advantage: neutral. Oddly enough, Bowser has a better fire breath than you. Use Rock Smash.
Captain Falcon, advantage: ??? You are a huge target for the knee. Wait, does the knee still do damage?
Diddy Kong, advantage: Charizard. Diddy has a problem knocking others off, as he has lousy knockback. You should have plenty of time to cook him. If the Diddy is all about banana spamorama, you can use n-air to do a lazy but effective approach.
Donkey Kong, advantage: ??? Not sure. Charizard will probably have his lights knocked out.
Falco, advantage: ??? I haven't tried this matchup, mainly because I fear Falco's speed. Charizard will probably end up using a lot of miserable d-smashes.
Fox, advantage: ??? See above.
Mr. G&W, advantage: Mr. G&W. G&W's b-air does silly silly silly amounts of damage to Charizard. Not to mention, it has sick knockback. Charizard's slowness will be his undoing.
Ganondorf, advantage: Charizard. Charizard can keep a nice distance from Ganon. Although Ganon has better arial attacks, Charizard's land game will give him the edge. Fire breath ftw.
Ice Climbers, advantage: neutral. ICs are fast and bothersome, but a lot of Charizard's attacks have a huge area of effect and can hit both of them. ICs can chaingrab Charizard easily.
Ike, advantage: neutral. Ike's sword will crush you. Fire breath is useful enough to balance this matchup. That, and Charizard can take a sword in the gut and keep fighting.
Ivysaur, advantage:
Jigglypuff, advantage:
King Dedede, advantage: King Dedede. Good luck approaching him. If you do get in range, don't let him grab you. If Dedede grabs you, you lose a stock. Better strategy: switch to Squirtle.
Kirby, advantage: neutral. Kirby's hammer is a serious threat in this matchup, because he can usually land it. Avoid his special attacks and you can counter with something hard hitting.
Link, advantage: neutral. Annoying matchup, but the Gale Boomerang will actually pull you closer to Link. Use glide to close distance and to avoid arrows. The clawshot will be hard to avoid if you don't have your aggression on.
Lucario, advantage: ??? I haven't played this matchup much.
Lucas, advantage: Lucas. Lucas's PK Thunder is ridiculously effective against large targets, since it can hit them multiple times. PK Fire will keep blasting Charizard back, and Psi Magnet will absorb the fire breath. Psi Magnet can also be used to knock Charizard off the stage.
Luigi, advantage: ??? Not sure, but I think Charizard has the advantage here.
Mario, advantage: ??? See above.
Marth, advantage: neutral. Charizard's fire breath will really give Marth a hard time, since he has no ranged attacks.
MetaKnight, advantage: ??? I play Charizard the least out of the three Pokemon, and as such, I don't know how a lot of matches feel. Someone else can fill the blanks.
Ness, advantage: Ness. I thought Charizard was supposed to be resistant to fire? Nope. PK Fire is hard to escape with Charizard. And then you eat the bat.
Peach, advantage: ??? I haven't seen this character since Melee.
Pikachu, advantage: Charizard. Charizard's reach and attack power beat Pikachu. Pikachu's down-b Thunder has potential to turn the tables of this match quickly; evade it.
Pikmin & Olimar, advantage: Olimar. Olimar plays too strong a ground game for Charizard to beat. Charizard is also a flying type Pokemon? It's a shame his arial attacks are slow and limited range. If Charizard and Olimar have some distance between them, Olimar will probably toss some Pikmin. If Charizard closes distance, Olimar will probably respond with his quick ranged grab or ranged smash. I have no idea how a Charizard can beat a defensive Olimar.
Pit, advantage: Pit. This character is stupidly powerful. Bow spam means you can't keep distance with Pit. If Charizard gets thrown by Pit, he'll likely be eating either a follow up u-air or a smash attack. Hyphen smashing u-air with Pit will wreck Charizard, even if he's on the ground. Oh, and he can reflect the fire. And recover from almost any situation. And his swords have higher priority than most of Charizard's attacks. And his bow attacks refresh all of his other attacks. Grah.
R.O.B., advantage: ??? I think ROB would ruin Charizard with gyros, but I'm not sure if Charizard could wreck ROB.
Samus, advantage: Samus. Samus's fat missles are the perfect thing to spam in this situation. That, and the charge-up plasma cannon. Charizard will have to close distance while dodging a barrage of projectiles, and every time he fails to dodge one, he gets flung back and has to try again.
Shiek, advantage: Shiek. Using only just Shiek's forward tilt A, you can rack up ~60% damage and finish with a u-smash to send Charizard off. Shiek's f-tilt is like Bullet Seed, except it hits whatever is in front of you and slowly lifts them. Shiek can also outrun Charizard and constantly restock needles.
Snake, advantage: ??? You can easily take out land mines with fire breath, but you'll probably be eating guided missles in the meanwhile. I don't know if this one is neutral or not.
Sonic, advantage: Charizard. Sonic gets roasted in this matchup. Charizard does well against other characters who have lousy priority in their attacks. Charizard doesn't need to land many hits to end this match. Sonic will have to constantly be running around, waiting to strike whenever an opportunity presents itself, all while baiting the enemy in to doing bad attacks.
Squirtle, advantage: Charizard. Rock smash is useless here. Squirtle can DI out of fire breath quickly. That's about all Squirtle has on Charizard. Squirtle has to deal a LOT of damage to Charizard to budge him, while all Charizard has to do is land a couple attacks in and Squirtle goes flying.
Toon Link, advantage: Toon Link. Projectile hell. TL's boomerang is evil, plus, some TL's use fancy techniques like double-arrow-short-hop and such. TL's b-air can really punish a Charizard, and everyone knows what his d-air is capable of.
Wario, advantage: Charizard. Wario is fat and floaty who is all about air control
Wolf, advantage: ??? I have no idea, but Wolf seems powerful enough to score a win here.
Yoshi, advantage: ??? Yoshi eats Charizard who turns in to an egg and falls to his doom.
Zelda, advantage: Zelda. She can easily get sweet-spotted kicks in on your dragon ***, and then punish your lack of fire resistance with ranged attacks. Think you can beat her in melee combat? Mind the sparkles. Oh hell, you got kicked again, trying to mind the sparkles.
Zero Suit Samus, advantage: ZSS. Only use Charizard against ZSS if your damage is already high (go go stock tank). You might be able to get some damage in with fire, but you're more likely going to be eating electric whip death.