Gen 3 imo has some of the best move animations in the entire series. No game before it, and no game after, has ever managed to match the epicness and rawness of Gen 3's move animations. The DS games, for as much as I love them, had move animations and sound effects that, while effective, were no where near as good as Gen 3's. Same applies to the 3DS games for that matter.
The move animations are simple but get the moves across well enough. But what further sells them is the amazing sound effects. Gen 3 only runs on a few sound effects, really, but on those few sound effects alone they really made many moves' animations that much more memorable. The animations are quick and simple, moreso than any generation before or after, and more than anything the sound effects add a whole layer of
power to the moves. Even if the move itself isn't strong, the animation+sound effect combination makes them feel powerful in a way that no other game has ever done since. Like with Thunder. My god, you can tell how strong that move is seeing those massive thunderbolts drop down with POWER and oomph. Or Hydro Pump.
Here's just about every animation from Gen 3:
So many good ones here. Even weak moves like Arm Thrust still have a lot of oomph to them because of the amazingly simple yet clean animation and sound effects they made here. It's simple, and runs off of a small set of sound effects, but the animations all work so damn well.
I can only imagine how awesome Close Combat, Focus Blast, Stone Edge, Hurricane, Play Rough, etc. would look in a Gen 3 aesthetic. Super simple, yet super clean animations and sound effects and Gen 3 really made a lot of moves have a ton of oomph to them, and no game since has ever succeeded in achieving the awesomeness that Gen 3's moves really had.