I've been using an android phone for a little over a year now. Here's a list of the apps i use or have used in the past.
Weatherbug or the weather channel - 2 best weather apps
barcode scanner - umm, scans barcodes
Shazam - Identifies songs using the microphone
last.fm, pandora - two great streaming music apps. Check out their websites first before trying to use on the phone.
stream furious - streaming audio app (shoutcast etc..)
gigbox - find concerts in your area
background - cool little background changing app
Astro - File manager app (delete, copy, rename files etc..)
cardar lite - bookmark your car location using GPS and find it later (great for nascar races)
Gameboid, BGCoid, Gearoid, Gensoid, Snesoid, Nesoid lite - emulators for older game systems (requires roms, if you know what those are, it should be easy to use, if not, then you'll need to do some research)
Dogcatcher - podcast app (not free)
google listen - free podcast app, but not as good
newsrob - rss reader
mileage - record your fill ups and calculate gas mileage
acar - similar to the above, but also tracks other maintenance and supports multiple vehicles (just started using this one)
notes - umm... take notes
shopSavvy - scan a barcode of an item and compare prices online before you buy in the store
finance - google finance app for stocks and such
remember the milk- Great todo list app. Works in conjunction with rememberthenilk.com (not free)
astrid - Similar to remember the milk. Great todo list that works with rememberthenilk.com
Locale - Share your location publicly via your phone's gps (not for the paranoid type)
Smart monitor - check how much cpu/ram/disk etc... is being used (both free and paid depending on features)
advanced task killer - keep tabs on which apps are running in the background and kill those suckers (helps with battery life)
By far, the apps I use the most now are: gmail and the newest maps version with turn by turn navigation. Unfortunately, I haven't found any killer games yet. Most are just OK. So far, I like robodefense and paralell kingdom (buggy, but interesting). Have fun and enjoy your new phone.