Racing games: PGR3 or 4, Forza 2, Dirt, Grid
Fighting games: Soul Caliber, Street Fighter on XBLA
You might ant to try Crackdown since it's similar to GTA/Saints Row style
Get a first party controller, they are cheap at Amazon.com, Wired controllers are cheaper than Wireless ones, If you buy a wireless one make sure you get a play and charge kit which allows to play while charging the battery, they are about $10-$15. Controllers can use regular batteries, i think it's AA maybe AAA. Use the ethernet cable that comes with the 360. Stand the console horizontal, I've heard that discs can get scratched if you have it vertical and don't move the console while the disc is spinning. There are a ton of free demos on XBLA, so try out the games before you buy them. Have fun.
