Luigi can be a difficult to handle at times. All of his specials have their uses. Many of which can actually lead to your doom pretty easily.
Fireballs are Luigi's spam. It can go about half of FD i think and floats, so it can be pretty dangerous and prevent Link from approaching with Zair at times. His fireball animation is actually really great to work in between combos, it gives him a nice float that can easily lead into any of his aerials, making comes like Fair>Fair>Fireball>Uair>Uair stuff possible. Fireballs can also be used to bait Link into air dodging and putting him into a bad place. SH fireball approach is pretty dangerous for Link, you shouldn't attempt to shield or dodge them, because a good Luigi can chase the fireball and grab or pivot grab after shield or dodge. Though on a good note i think Link has many attacks that can cancel or clank with Luigi's fireballs, someone should check that out.
Down Special is mostly use for recovery purposes, usually after the second jump but before the Up special.. Most Luigi players tend to fly straight up with it, and usually go above the ledge. If a Luigi does do this a lot, Link has a simple response with either just a Dthrown bomb at lower %s or Dair if you want to go for the kill. Ground based down special approaches are rather annoying to deal with. Overall it isn't too dangerous and doesn't do much damage. But Luigi can move really fast, and its probably impossibly to shield grab if a Luigi knows what hes doing. It can also lead to combos at nearly any % because of how little knock back it does. However it does require that Luigi stop near you to follow up, leaving him open to a shield grab or a fast OoS attack. Its probably not a good idea to attempt to dodge this attack as it has many hit boxes and Luigi could just sit on top of you and wait for your invincibility to run out. Though for the most part, all you really need to expect from this approach is just for him to move in and out for some quick damage and possible to knock you off balance.
The Green Missle is another special that can be used to recover. An attack based pretty much on luck for kills and many Luigi players will probably attempt to sneak a few in during the match hoping for a misfire. Though being as deadly as it is, it also is quite easy to predict where its going and counter with a multitude of attacks (use dair) lol.
Fire Punch is his final special. Another one of Luigi's deadly attacks that can kill at relatively low %s. It has a very small hit box and if he does miss you get a free smack at him with w/e you want to use. Most common ways of landing the fire punch is punishing a really slow attack such as a spin attack or grab. The other one is a close range jab cancel into fire punch. There is also a certain % in which his Nair can connect directly into his fire punch, but i dont think it can kill at that %.
Luigi tends to slide alot, and really far. Which generally makes it fairly impossible for Luigi to shield grab you unless you space incorrectly.
Most common things to worry about from Luigi is Utilts into aerials. This can easily combo Link at low %'s. Also with Luigi's huge recovery, he can easily to a deadly wall of pain off stage with Uair, Fair, and Bair. Its also important to note that Luigi's nair can easily get him out of many combos if you dont space correctly. Also that dair is capable of spiking.