Kirby vs. Pokemon Trainer
In general, you'll want to watch out for Kirby's main combo starters: Grab, U-tilt, U-air, D-air, D-tilt.
Kirby noobs will go for lots of grabs in hopes of getting the standard Kirby combo, F-throw to U-air. When you see the F-throw (slams you on your head going forward) get ready to Smash DI the U-air (away, or away & up). This will get you out of the U-tilt follow-up, especially if you jump afterward. If you see Kirby do the D-throw instead, you'll want to DI up and jump (if you DI up and toward Kirby you can try for a Footstool to get away) - you want to avoid the U-tilt follow-up after his D-throw. Kirby will only use a U-throw to KO, or to get you away from him quickly (such as in Doubles). Kirby's B-throw is pretty bad and will be rarely seen.
Kirby's U-tilt hits close to him, so you'll want to look out for that when he's next to you, especially when he's facing away from you. It will hit you into a juggle so that heavier/fast fallers can get caught by multiple U-tilts at low percent (such as Charizard). Kirby can usually get two U-tilts on someone who's at 0% even if they're a floaty character. Follow-ups to U-tilt include B-air, U-air, N-air. If you try to Air Dodge or attack through the follow-up, Kirby can change things up by going with punishment or counter attacks, such as Grabbing you after your Air Dodge. You'll want to DI (or Smash DI) the U-tilt horizontally, if you need to get away even more, jump away so that you can avoid his "punishment" attacks.
Kirby's U-air hits on Frame 10 - it's not fast, but it's not very slow either. What makes up for it's mediocre speed is the range it has over Kirby's head. It hits pretty high, so don't underestimate the range Kirby has when he's under you. The U-air also hits in front of Kirby when it starts up, and drops down behind him as it finishes. He can use the first part as an approach, or he can use it while facing away from you to hit with the finishing frames and drop down into an Auto-Cancel'd follow-up. The best follow-up for the U-air at lower percents is a U-tilt, which I went over already. At very low percents, a U-air can lead into a grab as well as a U-tilt, F-tilt, Jab, or D-tilt, depending on the weight of the character. For the most part, U-air is used as a launcher and does 10%.
Kirby's D-air is versatile. When the opponent is on the ground, Kirby can drill into them with a multi-hit attack that builds up damage and then pushes the opponent away when he hits the ground. The 'ground hit' of the D-air is weak enough that Kirby can follow up with a Grab, Dash Attack, or F-smash. Most of the time, the Grab is the best thing he can use to follow-up since the Dash Attack and the F-smash gives the opponent enough time to get a Shield up. Since the opponent will want to avoid getting a F-smash, they'll be holding the Shield button, so again, the Grab is preferred. Sometimes, though, the D-air can trip, or even launch the opponent slightly into the air where they will spin around, either of these is an excellent set-up for an F-smash or U-smash. A Dash Attack follow-up after a normal D-air is to punish people trying to dodge a grab.
Off-stage, the D-air can be used to gimp recoveries. All the Pokemon get gimped pretty hard if they ever get hit by the D-air. The knockback isn't tremendous, but it drags the victim downward with a multi-hitting attack which makes it hard to avoid and difficult to recover from. The good news is the D-air takes 18 frames of start-up, so you can usually avoid it when you see it coming, or you can try to hit Kirby first. It's priority isn't great, either, so you might be able to hit Kirby out of it, just don't get hit by it when trying.
Kirby's D-tilt is very quick, and he'll squish down when using it making him even harder to hit than with his normal crouch (which is already pretty small). There's not much you can do against the D-tilt since the range and speed is good enough to avoid punishment, Kirby's stance is defensive, and it can shield stab taller characters when their shield is low. When it hits from 0% to 150+%, it can trip. What Kirby does to you after the trip depends on how good the Kirby is and what he wants to do to you. If he wants to build damage, he'll usually go for a grab, but skilled Kirby players can also try for a reverse U-tilt to juggle you. If you're at higher damage, you're going to get F-smashed. If you tripped close to Kirby, the F-smash is pretty unavoidable, but if you're farther away you might be able to roll away before the F-smash reaches you, or at the very least it might not sweet spot you. Kirby can also go for a Dashing Grab, Dash Attack, F-tilt, or another D-tilt. The F-tilt and D-tilt after a D-tilt are usually done as a safe combo - the F-tilt to knock you away, the D-tilt is best used when the first D-tilt didn't trip and the follow-up D-tilt might.
Off-stage, Kirby has a pretty easy time vs. Pokemon Trainer. He's has plenty of jumps as well as F-special and U-special to add to his recovery. He can use D-special to drop back onto the stage while practically invincible and get all his jumps back, as well as a chance to hit anyone below him for 18% vertical damage.
When edge guarding, Kirby will usually jump out with B-air, F-air, D-air or F-special to intercept any of the Pokemon's recovery, except for Ivysaur. Against Ivysaur everyone knows to just edgehog. Charizard probably has the easiest time against Kirby when recovering since he has multiple jumps, plus he's a heavy weight so he can DI upward and try to Glide back to the center of the stage before avoiding Kirby's offense.
Charizard vs. Kirby
Kirby will be trying to rack up damage with combos since Charizard is easy to knock around. Charizard's heavy weight won't help much since Kirby will either KO around 100% to 120% with his F-smash, or will go for off-stage KOs. If you can avoid getting hit by F-smash, Hammer, and Stone, Charizard should be able to survive for a good amount of damage. Build up damage with Flamethrower, Rock Smash, and Grabs. When Kirby is at a good damage, KO with D-tilt or U-smash, go for a D-throw when he's around at higher percents. Watch out for Kirby's close quarters fighting, he's fast, builds up damage, and can get you off-stage quickly. Kirby's Charizard hat can be used against any Pokemon to build up damage and give Kirby a good spacing attack, don't let him get it!
55-45, Kirby's favor.
Ivysaur vs. Kirby
You'll want to use your range to its fullest, do not let Kirby near you. Of course, he'll be trying to do the opposite, so when he does get near you either avoid him completely or try to get a quick hit in that does not leave yourself open. D-tilt, and the multi-jab are some of your better choices when up close, but their purpose is to put space betwen you and Kirby. If you can get a Grab/Pivot Grab on Kirby, toss him off stage, or just throw him so that he's not next to you anymore. F-air and U-special can KO Kirby decently, or better yet they can knock him far enough away to switch out to a better Pokemon. F-smash is a great KO attack on Kirby, the range and power are great to finish him off and switch out. Don't get knocked off stage. Kirby's Ivysaur hat is bad news, don't let him get it!
65-35, Kirby's favor.
Kirby vs. Squirtle
I think Squirtle is superior to Kirby with attacks in every way... except for range. Kirby's B-air is just a little bit bigger and easier to use, so expect him to spam this. Also, Kirby's F-smash is an awesome KO attack against Squirtle, it has the range, priority and power to take out Squirtle easily, and this is perhaps the biggest threat. Off-stage, Kirby also has the advantage since he has multiple jumps and can gimp Squirtle fairly easily. What you'll want to do is stay on-stage, stay mobile, and get the hits in. Stick with fast, defense attacks such as retreating B-airs and fading F-airs. Getting Kirby into a juggle might be difficult, but you can build up some good damage if you do. Getting the KO on Kirby will be hard since hitting with a U-smash might be predicted too easily, so you'll mostly be looking to get a D-throw on him which will KO Kirby under 120% to 150%.
55-45, Kirby's favor.