Cool to see another up and coming pikachu main!
As with a lot of characters in Melee, there aren't really many 'bread and butter' pikachu combos, but there is plenty you can look for. While you can't always follow up directly, keeping the opponent above you is almost always beneficial, and pikachu's techchase game is also fairly strong. I'll divide up opponents into non-floaty, medium, and floaties.
Non-floaties (Fox, Falco, C.Falcon):
These will be the most easy to combo. Uair juggles really well, and will often lead to a number of things. Rising uair leads into more uairs, and then other aerials at higher percents. You can also get grab out of uair potentially, which is powerful. Usmash combos into aerials or another usmash at low-mid percents. Uthrow is a chaingrab on FD until around 100%, or whenever you don't have a platform above you.
You usually do want to end these combos with something that will send the opponent offstage. This isn't entirely because they are fastfallers, but because the recoveries of the 3 fastfallers are often punishable (and they are matchups where you reallllly need to optimize your punishes).
Medium (ex. Sheik, Mario, etc.):
This covers the majority of the cast. You won't have as brutal of juggles as you would on fastfallers, but you can still do a lot with your combos. The main thing you're looking for while comboing these characters is for them to use their second jump. Once that happens, follow and uair spike them offstage if possible.
Since you can't juggle as hard, you have to decide (or recognize, depending on your position) if you want to end your combo with a strong aerial (dair, nair, bair), end with a uair spike to put them in an edgeguard position, or end with first-hitbox uair to send them above you and see if they unsafely use their second jump.
The only real common-combos on these characters i can think of are rising uair -> nair/bair (or dair at low percents), and uthrow quick combos. At super low percents, you can get usmash out of uthrow on some characters (ex. Ganon, Sheik). Others, you can get dair out of it. At higher percents you can at least get a nair or uair, though in this case it may be better to just bthrow them off stage (if that's an option) instead of uthrow.
Lastly, running nair->usmash can work if you hit a little late with your nair. This works at most percents, but you generally have to adjust when in your nair arc you hit them depending on their percent (the higher their percent, the sooner you can hit them) and may have to wait very slightly after you land and or dash forward slightly depending on their DI.
Floaties (ex. Peach, Jiggs)
You can't combo floaties, except at very low percents kinda. They're annoying. Either try to chain a couple nairs or move them around with uair to wherever gives you a positional advantage. The nair->usmash thing can work at lower percents, so it's worth going for.
There's more that I probably haven't thought of (or is too situational/insignificant to mention). Hope that helps a bit!
