Hello, First, Welcome to Amiga.org! Second, yes the A1200 would be the best bet for games. You will have more options if you expand the RAM, though.....best part is you can install WHDLoad, which lets you run older, floppy-only games from the HD, as krize hinted at above.
Funny, you're an Atari fan trying to get an Amiga, I'm an Amiga fan looking to get an Atari!
Again, welcome. :-D
