Ok, well this is the first non-nintendo console I've owned EVER so I need help finding a game. (Was given a 360 for Christmas by surprise with no games.)  Any game idea is good just name MUST HAVES and possibly deals if you know of one.

Must Haves!!;
Mass Effect, Forza 2, Gears 1 & 2, Lost Odyessy, ToV, Fable 2, GTA4

Good Deals (Cheap Now);
Kameo, Banjo, Dead Rising, Crackdown, Bioshock

Good Acrade Games;
Braid, Duke Nukem 3D

Enjoy your 360 and get some of the games i've suggested....

Edit: Oh and any genre except sports.

Lol, I was about to ask your favorite genres.

Halo 3 is just a classic goodie.  Orange Box, Gears of War 1 and 2, Call of Duty 4 or 5, all good FPS's.  Dead Space is a good horror/shooter.  Lost Odyssey, Fable 1 and 2, Mass Effect, Fallout 3 = all great RPGs!  Some more arcade titles: Castle Crashers, Portal: Still Alive, Geometry Wars

Get Gears 1 and Halo 3 you should get them cheap enough second hand.
I hope you enjoy the 360.

Halo 3
Mass Effect
Gears 1 & 2
Call of Duty 4 & World at War
Banjo Kazooie
Viva Pinata 1, 2 & Party Animals
LIPS
Guitar Hero: World Tour

Good games are

Halo 3
Gears 1 and 2
COD 4 and 5
Bioshock
Mass Effect
Ninja Gaiden 2
GTA4
Fable II
Kameo
Dead Rising
Lost Odyssey
Crack Down

Banjo Kazooie

Dead Space

Fallout 3

GEOMETRY WARS 2

SRSLY

Take a look at mass efect , fable 2 and fallout 3.

The 360 is the best when it comes to either WRPGs or JRPG's so take advantage of that.

All these people should hang their heads in shame because no one listed the God of all games:

Oblivion