I use the radio/music apps a lot. Pandora and AllRadio get the most use, and I listen to XM/Sirius Satellite Radio on WunderRadio. SimplifyMedia is pretty amazing as it allows you to listen to your computer iTunes Library on your iPhone from anywhere.
Slingbox has become a staple--great for watching satellite TV anywhere in a wifi zone. StockWatch is the best portfolio monitor I've found, and CarStat keeps track of gasoline and other car-related expenses.
AIM did a good job of porting there software to the iPhone, and The Weather Channel, with its radar maps is my favorite for keeping me dry. I also love Truphone for cheap to free calls using wifi, especially for international calls. It's also a great minute saver for domestic calls to people you can tell to get on Truphone with their smartphone or computer.
Remote, used with Airport Express(es) lets you control streaming music from iTunes to sound systems anywhere in your house. You can adjust the volume, change the music, or move the sound from one part of the house to another, all using just the iPhone.
I've also become a devotee of eReader. It's nice when you're stuck somewhere, waiting for whatever, to be able to pull a book out of your pocket.
It's all good, and I'm sure you'll discover your tastes and needs. The only other advice I could pass on is to avoid the "novelty" apps. A lot of them are very clever, and very cool--once. Maybe twice. But I had to admit that turning my iPhone into a Rainstick kind of got old fast. Not to mention the cow tossing and the bubble-wrap popper, and I am rapidly embarrassing myself....