Yoshis Island I find to be the best neutral overall for Pokemon Trainer. Battlefield is another good one too, although like Retro stated, it can be difficult to manage if Charizard gets thrown or knocked above the platforms. I tend to abuse more than be abused on that stage though, so I'm usually fond of it.
Lylat Cruise is good for Charizard, I'm not fond of it for anyone else really. I generally dislike the stage.
Oddly enough, I found Rainbow Cruise to work very well. Especially for Squirtle. I thought that Ivysaurs weaknesses would be heavily exploited here, but that doesn't seem to be the case most of the time I've played.
Other stages I love to Counterpick, barring they're legal, are Norfair, Corneria and Pirate Ship. Norfair is a fantastic Ivysaur stage. Nair/Bair combos everywhere, and you can maneuver around the stage really well. Squirtle does well here too. Corneria I discovered is good for Charizard and Squirtle, mostly due to the very low ceiling and the brutal stage spikes you can perform on against the wall of the fin with throws. Pirate Ship... well. If you're Charizard, and they go into the water. You win.
Edit: How would you guys find other Melee stages, such as Brinstar, PS1 and Dreamland?