I just picked up a 360 Elite and Gears of War a little over a week ago.

 I'm somewhat regretting it though as I'm not really very good at the Gears multiplayer.

I'm going to sound like a bit of a bandwagon, but for PC it was probably the original Quake (more specifically QuakeWorld) that really got me into it. Trust me, there's nothing quite like running around in an axe-only game!

 For consoles, Final Fantasy 7 on the original PlayStation.