There are two types of Amiga gaming:
There's classic Amiga gaming, the older stuff from when the Amiga was more popular.
Then there's the more recent stuff that'll require a PPC accelerator, 3D accelerator, etc.
I assume you're after a machine for the former :-)
An A1200 should suit you pretty well, although you should make sure it's a Commodore model, not an Amiga Technologies model, as those have trouble with bootable floppy games (copy protection issue). An Amiga Tech model should be fine if the floppy's been replaced.
A 200MB hard drive sounds like it'll be enough for what you're planning to do.
You'll probably want an 030 accelerator with some RAM. You should be able to get by with less than 16MB.
This'll probably go for $200 US on ebay. You can get it cheaper in a direct sale.
You could also get an A500, but you won't be able to run AGA games. It's also harder to expand (based on the availability of hardware at present). It's considerably cheaper, though.
Welcome to the Amiga!
If you need anything else, don't hesitate to ask!