Amiga 500 is nice with older Amiga games (most of the games), but for some later games (1992 and onward) A1200 is probably the best choice.
Older games used OCS/ECS chipsets, and usually they are playble after some fiddling with all Amigas. When Amiga 1200/4000/CD32 (which used AGA) were released, some games mainly for those machines became available.
Using software called WHDLoad, you can install most (even those older) games to harddrive, which could make A1200/A4000 perfect for Amiga gaming.
I don't have links available right now, but there are many sites listing Amiga games and their requirements. You could also try out games using some Amiga emulator, like WinUAE before getting in the process to get them running on the real thing.
Keep an eye out for Minimig too, which is small motherboard emulating Amiga 500 nearly perfectly with classic joystick ports running games from flash cards.
