This depends on what you consider an advanced technique. Honestly, things like good spacing and timing or matchup knowledge matter about a billion times more than any AT so if you're just looking for how to improve, that's the best place to start.
However, there are a few things you might want to pick up.
B-reversals are huge, especially for characters with directional up specials (Squirtle, Kirby, Ike, Meta Knight, Marth, etc.). Even if you aren't using one of those characters, it's still something you need to be able to do.
Glide tossing is not quite universal (Yoshi can't do it), and a fair number of characters get a zero or extremely low distance out of it. However, it's super big for a few characters like Diddy Kong and generally worth learning if you use anyone with a useful glide toss distance. Just be aware that a few characters (like Mr. Game & Watch) require ridiculously good timing if you want to get a useful distance.
DACUS (dash attack cancel up smash) is huge for everyone who gets a useful distance off it. That AT thread refers to it as "Boosting Technique: Smash-Boosting (a.k.a. Snake Dashing, etc)" and I think it has 5 or 6 other names, but it's worth using if you use someone who gets a good distance (Snake being the poster boy). Your icon shows Wario; he gets something useful out of it.
Optimum survival when hit is ATish, and it's something everyone needs to understand. It kinda goes like this...
SDI? SDI is really hard to use and has a pretty small benefit for most moves, but with multi-hit moves or in very specific circumstances otherwise (like when hit by Falco's down aerial at the end of a chaingrab), it's very useful. The best way to SDI is to wiggle the control stick between two adjacent directions including diagonals (like wiggle between up and up-right). If hit with a single powerful hit that will quite possibly kill you, don't bother with this since you are going to have to input a bunch of other more imporant stuff.
DI? Regular old DI is just holding a direction the frame you exit hitlag. This changes your trajectory but not your launch speed; your ultimate goal is to direct yourself to fly into a corner since the stages all have rectangular blast zones. This generally means holding the stick perpendicular to the launch direction, NOT the intuitive holding it toward the stage (though do hold toward the stage after you are already flying so you air control that way as soon as you can).
Do an aerial? You can do any aerial (or airdodge or throw an item) halfway through hitstun in Brawl. This doesn't directly help you, but you do get the ability to do other moves once you finish, and you can begin to fastfall while doing aerials or when you can do other moves which can help you survive vertical hits. The general rule is to throw an item if you are holding one but otherwise do your fastest aerial (fastest being whatever finishes fastest, not hits fastest). Airdodging is tempting, but it's not a good option and doing it when hit is a bad habit. Also, don't think that doing momentum altering aerials (like Toon Link's down aerial) is going to slow you down; it doesn't, and most of those take a very long time to finish so they're actually terrible options.
Momentum cancel? Certain special moves slow down the user when used while launched. The most dramatic and useful case is with Mr. Game & Watch's non-filled bucket; it stops him completely (DK's up special and Yoshi's forward special do this too, but they're not so useful for obvious reasons). Quite a few if not most characters have some special that at least somewhat slows them down; you'll have to explore.
Double jump? If you don't have a good way to momentum cancel and are flying horizontally, your general next best attempt is to double jump toward the stage. This does give you a boost AWAY from the stage which sucks and sometimes kills you, but in general it helps you live more than it hurts you. Just don't do it when you're already in the bubble offscreen.
I didn't mention it in the rest of that, but obviously be ready to tech if you're hit into a surface. Being good at teching is really important to surviving on some stages (and will be super helpful to you in not getting owned by Mr. Game & Watch's down throw). Actually, it's not really an AT, but you absolutely need to be able to tech Mr. Game & Watch's down throw 100% of the time.
Optimum grab breaking is also something you should know how to do (extremely important for Mario, Luigi, Samus, Ness, and Lucas but generally useful for everyone). The best technique is to rotate the control stick while mashing as many buttons on the right face of the controller as you can (I am not yet completely sure what the absolute best right side strategy is, but the control stick is the really important part). This severely damages your ability to DI throws so be smart about it, but it really cuts down on the number of pummels your opponent can get away with.
This probably isn't the first thing to occur to you when learning your character, but be sure you know your best answers to ledge stalling before it comes up in a tournament. If you use Meta Knight or Mr. Game & Watch, you should practice ledge stalling yourself.
Speaking of "best move to use when", you need to know your best move to challenge the Mach Tornado (if you have one that can) and your best move to get pikmin off you in general. You should be aware of which of your moves have elemental properties (if any); they matter against Olimar and Pokemon Trainer.
If you use Ice Climbers, you need to learn how to do their infinites. You can pick your variant, but people should not generally survive being grabbed.
If you are using a character who can force jump breaks on flat ground (Peach, Bowser, DK, Yoshi, Wario, Zelda, Sheik, Ganondorf, Zero Suit Samus, Captain Falcon, Ivysaur, Marth, Ike, Snake), you would benefit from learning what free stuff you get against the characters with bad jump breaks (the main three to abuse are Wario, Meta Knight, and Squirtle [I think but am not sure that Lucas is a general 4th] but you can sometimes get stuff against other characters). If you use Bowser or Yoshi in particular, you should really look into grab breaks a lot since they're a really big part of their games. If you use Marth or Charizard, be prepared for the grab release chaingrab on Ness and Lucas.
Falco, Pikachu, and King Dedede have some really major chaingrabs that you need to learn if you want to be competitive with them. You should know where your character stands in terms of being chaingrabbed by all three regardless of who you main.
If your character has a jab lock (or a similar move that does the same thing), you really should learn it. I don't think there's a really good comprehensive list of which moves have this property anywhere, unfortunately. Do be aware that some jab locks (like Olimar's) are so difficult that they are probably not worth the effort to learn.
There may be more character specific stuff you need to know; the character boards will hook you up with that. You want to have some degree of character knowledge of the whole cast and stage knowledge of every stage that's legal in your region so be sure to take full advantage of friendlies for that even if it puts you outside of your comfort zone. Using random low tier characters on stages you don't like may seem like a bad idea, but you will be thankful for every little thing you learn when someone you've never met picks that character and that stage in a tournament.
This turned out mega long and into a general "things to look into" post, but it should have some good stuff in it.
However, there are a few things you might want to pick up.
B-reversals are huge, especially for characters with directional up specials (Squirtle, Kirby, Ike, Meta Knight, Marth, etc.). Even if you aren't using one of those characters, it's still something you need to be able to do.
Glide tossing is not quite universal (Yoshi can't do it), and a fair number of characters get a zero or extremely low distance out of it. However, it's super big for a few characters like Diddy Kong and generally worth learning if you use anyone with a useful glide toss distance. Just be aware that a few characters (like Mr. Game & Watch) require ridiculously good timing if you want to get a useful distance.
DACUS (dash attack cancel up smash) is huge for everyone who gets a useful distance off it. That AT thread refers to it as "Boosting Technique: Smash-Boosting (a.k.a. Snake Dashing, etc)" and I think it has 5 or 6 other names, but it's worth using if you use someone who gets a good distance (Snake being the poster boy). Your icon shows Wario; he gets something useful out of it.
Optimum survival when hit is ATish, and it's something everyone needs to understand. It kinda goes like this...
SDI? SDI is really hard to use and has a pretty small benefit for most moves, but with multi-hit moves or in very specific circumstances otherwise (like when hit by Falco's down aerial at the end of a chaingrab), it's very useful. The best way to SDI is to wiggle the control stick between two adjacent directions including diagonals (like wiggle between up and up-right). If hit with a single powerful hit that will quite possibly kill you, don't bother with this since you are going to have to input a bunch of other more imporant stuff.
DI? Regular old DI is just holding a direction the frame you exit hitlag. This changes your trajectory but not your launch speed; your ultimate goal is to direct yourself to fly into a corner since the stages all have rectangular blast zones. This generally means holding the stick perpendicular to the launch direction, NOT the intuitive holding it toward the stage (though do hold toward the stage after you are already flying so you air control that way as soon as you can).
Do an aerial? You can do any aerial (or airdodge or throw an item) halfway through hitstun in Brawl. This doesn't directly help you, but you do get the ability to do other moves once you finish, and you can begin to fastfall while doing aerials or when you can do other moves which can help you survive vertical hits. The general rule is to throw an item if you are holding one but otherwise do your fastest aerial (fastest being whatever finishes fastest, not hits fastest). Airdodging is tempting, but it's not a good option and doing it when hit is a bad habit. Also, don't think that doing momentum altering aerials (like Toon Link's down aerial) is going to slow you down; it doesn't, and most of those take a very long time to finish so they're actually terrible options.
Momentum cancel? Certain special moves slow down the user when used while launched. The most dramatic and useful case is with Mr. Game & Watch's non-filled bucket; it stops him completely (DK's up special and Yoshi's forward special do this too, but they're not so useful for obvious reasons). Quite a few if not most characters have some special that at least somewhat slows them down; you'll have to explore.
Double jump? If you don't have a good way to momentum cancel and are flying horizontally, your general next best attempt is to double jump toward the stage. This does give you a boost AWAY from the stage which sucks and sometimes kills you, but in general it helps you live more than it hurts you. Just don't do it when you're already in the bubble offscreen.
I didn't mention it in the rest of that, but obviously be ready to tech if you're hit into a surface. Being good at teching is really important to surviving on some stages (and will be super helpful to you in not getting owned by Mr. Game & Watch's down throw). Actually, it's not really an AT, but you absolutely need to be able to tech Mr. Game & Watch's down throw 100% of the time.
Optimum grab breaking is also something you should know how to do (extremely important for Mario, Luigi, Samus, Ness, and Lucas but generally useful for everyone). The best technique is to rotate the control stick while mashing as many buttons on the right face of the controller as you can (I am not yet completely sure what the absolute best right side strategy is, but the control stick is the really important part). This severely damages your ability to DI throws so be smart about it, but it really cuts down on the number of pummels your opponent can get away with.
This probably isn't the first thing to occur to you when learning your character, but be sure you know your best answers to ledge stalling before it comes up in a tournament. If you use Meta Knight or Mr. Game & Watch, you should practice ledge stalling yourself.
Speaking of "best move to use when", you need to know your best move to challenge the Mach Tornado (if you have one that can) and your best move to get pikmin off you in general. You should be aware of which of your moves have elemental properties (if any); they matter against Olimar and Pokemon Trainer.
If you use Ice Climbers, you need to learn how to do their infinites. You can pick your variant, but people should not generally survive being grabbed.
If you are using a character who can force jump breaks on flat ground (Peach, Bowser, DK, Yoshi, Wario, Zelda, Sheik, Ganondorf, Zero Suit Samus, Captain Falcon, Ivysaur, Marth, Ike, Snake), you would benefit from learning what free stuff you get against the characters with bad jump breaks (the main three to abuse are Wario, Meta Knight, and Squirtle [I think but am not sure that Lucas is a general 4th] but you can sometimes get stuff against other characters). If you use Bowser or Yoshi in particular, you should really look into grab breaks a lot since they're a really big part of their games. If you use Marth or Charizard, be prepared for the grab release chaingrab on Ness and Lucas.
Falco, Pikachu, and King Dedede have some really major chaingrabs that you need to learn if you want to be competitive with them. You should know where your character stands in terms of being chaingrabbed by all three regardless of who you main.
If your character has a jab lock (or a similar move that does the same thing), you really should learn it. I don't think there's a really good comprehensive list of which moves have this property anywhere, unfortunately. Do be aware that some jab locks (like Olimar's) are so difficult that they are probably not worth the effort to learn.
There may be more character specific stuff you need to know; the character boards will hook you up with that. You want to have some degree of character knowledge of the whole cast and stage knowledge of every stage that's legal in your region so be sure to take full advantage of friendlies for that even if it puts you outside of your comfort zone. Using random low tier characters on stages you don't like may seem like a bad idea, but you will be thankful for every little thing you learn when someone you've never met picks that character and that stage in a tournament.
This turned out mega long and into a general "things to look into" post, but it should have some good stuff in it.
