I've been waiting for a thread like this. Luigi can 1. Rack up damage quickly in the air with excellent aerial strings or juggles. 2. Kill at considerably low %. If you do it right, Luigi shouldn't need to approach all that much. Basically, an almost 0-60% damage string first, second approach setup for firejumppunch(midrange jab to firejumppunch on most characters, dtilt trip and then quickly step over into firejumppunch on small characters, and I'm sure there are a few more), or 0-60% damage string into an edgeguard, or finally a 0-60% string into a second damage string w/ an fsmash or upsmash squeezed in as soon as the % reaches that level. Finally, if you've gotten 2 rounds of damage in but no KO yet, you get your last approach w/ your easiest to land kill options available to you(sweetspotted sex kick and backthrow).
So step number 1 is limit the amount you have to do approaching as much as possible.
Luigi should have to put a noticeably more effort into approaching vs. other aspects of his game. He doesn't have the range of some of the other characters, and he doesn't have the moveset to force others to approach him. Step number 2 is having a variety in approaching-
First- projectiles. At any decent range, you can airdodge projectiles pretty safely. However, don't get into the habit of doing this close up, because it'll more than likely leave you vulnerable. However, you may find that trying a shielded approach on the grounded constantly gets Luigi pushed back because of his low traction. Since rolling can be predictable and punishable, powershielding and spotdodging projectiles are you safest bet. Luigi's spotdodge is of high quality and is quite safe to use.
After that, there is usually a certain range where you're outside of any immediate ground attack(shield-dash grab, dash grab, dash attack, and short-hopped aerials excluded), and too close for projectiles. From here, against the majority of characters, you simply need to "mindgame" you're way in for your approach. Either A- they make a mistake and you can punish a laggy attack(rolling is very effective here) or...
RAR a bair. Luigi's bair is possibly his highest priority move, and as an aerial that means it has great range/extended hitbox to it. I disagree with saving Bair for a KO move, Luigi already has enough other options for KOing, but not for approach. His Dair and sex kick can even KO in the air if you want, and bair will still be useful for edgeguarding and w/ reduced knockback KOing when they're already off the side.
Shield-dash. If you shield dash and actually powershield an attack, you're finally close enough for a number of different options, including grabs.
Any number of short-hopped aerials or double aerials. When you use this method, you have to use Luigi's aerial mobility to do something like fake them out first, then move in to hit, or hit with one aerial that they shield and then as they go to shield grab you, to hit with the second, or move away and land just outside their reach, etc. Note on when using dair- I forget how (I'd need a controller in front of me to actually say how) but using it certain ways auto-fastfalls you, which can be good or bad depending.
Fireball. You're fireballs are fast enough that at the range that most projectile spammers can't reliably projectile spam against you but you are just outside their standing ground attack range, you're fireballs dominate. A fireball can disrupt or simply mess-up a person to get your opportunity in. It's very useful.
For example, I really like doing a fireball to a tornado. If the fireball hits or is shielded, the tornado usually is fast enough to always reliably hit the fireball-hit person, or the tornado can move onto the shield and keep hitting them until it gets a shield poke in and pops them up in the air on the last hit. But there are many other options too following a fireball, including more fireballs against those that don't have reflectors.
And of course the tornado itself is a quality approach. It's fast, can be surprising, and as long as you have good control on it to make sure that if you got one hit in you also hit with the last hit to pop them upwards with it, you're good. The Tornado can get a lot of use on approaching and chasing, it's a pretty unique approach option that a lot of characters don't share.
Another option is if they're again, at that certain range, they may try to get you with a dash attack or a short-hopped aerial. Both of which can be shielded and shield-grabbed OR shielded and then firejumppunched out of shield. This is ideal, because that's you're best killing move and you're best setup for aerial combos too.
Lastly, over-B. Yes, over-B. If you use this on rare occasion and by charging it different lengths, you'll be surprised how often it hits people. When it hits a shield, it usually sends them sliding out of range, if they dodge or jump over it, they tend to be too far away to punish the lag, and since it can come at surprising speed(especially if misfired) and at any time you release, it can even hit them. And if it hits them, you get extra points for making them feel very very stupid.
Some characters out there are just going to be able to spam you with projectiles and outrange you. For example, Zelda. The trick is to keep your approach unpredictable by keeping it varied and set yourself up for excellent damage strings or KOs, and master the ability of punishing mistakes. I didn't mention the dash attack because I don't think there is anytime a dash attack is more reliable or useful than an upsmash out of a dash unless you're going to be catching an item.