Well halo (You can grab 1, 2 and ODST from games on demand, play through them then get 3 and Reach separately)
Probably the best introduction to 360 gaming, given the ease of control, the new features in each game and the rich storyline.
Gears of war is also a good one to pick up for the awesome combat and fun gameplay mechanics, Both games are quite cheap right now, so you can probably pick them up preowned for next to nothing
Another good purchase is Red dead redemption, a bit more expensive, but certainly worth a look.
And since the 360 is the platform for racing games you need to get: *takes breath*
PGR3
PGR4
Blur
The Club (okay not strictly a racing game but made by bizarre)
Forza motorsport 2
Forza motorsport 3
Moto GP (whatever year the latest game came out)
Flatout UC
Split/Second
Test Drive Unlimited
Test Drive unlimited 2
And possibly a 360 racing wheel, either the official one or a madcatz one, both are quite good.
If you can only get a few of those DO NOT miss out on Forza 3 or PGR4, seriously, those games kick so much ass.
And if you DO get forza 3, pony up for the ultimate collection, that adds about another 120 hours worth of new and original fun to be had with the game, Turn 10 are the greatest at racing games, tons of customisability packed in over the most impressive driving physics in any game, forza is one of those games that should be in any 360 owners collection just for the sheer brilliance of the gameplay.
Other than that, all the exclusives are certainly worth a look when you have the money.