I also was not blown away by GTAIV. I don't mean to imply it is a bad game, it is in fact a fantastic game BUT I think I expected more from it. Other than visually, the size of the city and no load screens it isn't much different than the previous GTA games. I enjoyed San Andreas a bit more. I haven't fininshed GTAIV, I just never really feel like playing it.

Again, I am NOT saying it is a bad game in any way I just had higher expectations for it. San Andreas had some very cool new things, all the different cities and the land between them, complete with little towns. There was nothing really mind blowing and new in GTAIV. It does have a deeper story but having to keep all your contacts happy is annoying. The mini-games like darts and bowling are fun the first time but it is tedious to have to keep taking your friends out on man-dates.

What types of games do you like? There are some great ones in most genres for the 360 with shooters being very well represented.

Since you never owned a 360 before there are a lot of older titles that you will be able to get cheap but will be new to you.

Dead Rising is great. Basically you get to play through Dawn of the Dead, the entire shopping mall is full of zombies and pretty much everything from any store is a weapon.

Oblivion- I can't think of too many games that will give you as much bang for your money.

Saint's Row- it isn't GTAIV but you can likely get it for $20 or less.

Gears of War-looks incredible in hi-def.

Fight Night Round 3 best boxing game ever.

There are a lot of good racing games. Forza 2 is probably the front runner but I enjoyed Need for Speed; Most Wanted a lot more, Burnout Paradise is a lot of fun too.

Bioshock-incredible game, exellent story and great atmosphere.

Crackdown-didn't get a lot of acclaim but I had a blast playing it. You basically get to be a super-powered cop and go to war with bad guys across an entire city.

The two best online games in my opinion are Frontlines and Battlefield Bad Company, now these are both team oriented games , if you prefer deathmatch style stuff than go with COD4 (very good single player campaign but it is VERY short, if you don't play online just rent it) or the ever-popular Halo3 (I didn't care for it much but I'm in the minority on that).

Congrats on your decision, you are going to find a LOT of games to choose from (a lot more than for the Wii or PS3) use resources like Gamespot to check out people's opinions so you don't waste your money. There are so many really good games for the 360 that it would be a shame if you wasted money on some bad games.