If you don't mind a possibly-higher budget and tinkering with computers, you can upgrade a desktop into a gaming PC for about $200-$300. This is the much better system for FPS's and RTS's. Also allows for lots of modding, etc.
Xbox 360 is sort of the "mainstream hardcore" platform. It's got Halo 3, Gears of War 1 and 2, as well as Xbox Live. You need to pay Microsoft to host or join any game servers.
PS3, like the PC, has free multiplayer. It has many of the same games, and some exclusives like MGS4. You can install Linux on it. It lets you play BLU-RAY movies as well.
The Wii is for a more casual player, and works well if you have other people in the household that want to join in without becoming an expert at sniping and grenade jumping. Be aware the vast majority of Wii games are shitty knockoffs, all trying to imitate a very solid bunch of games put out by Nintendo, many of which earn scores in the 90s.
That's sort of a brief primer for you.