I'm a new Droid owner too. Here are some of my favs:
SportsTap - Tell it what teams you like and it will alert you with scores, also an app to check scores/rankings/schedules/etc for most sports
WeatherBug - Puts the local temperature in your upper left menubar and sends severe weather alerts, also a great weather app
OI File Manager - Lets you browse the file system and perform common file operations
ConnectBot - an SSH client
Pandora & Slacker - Personalized Internet Radio Players
PDANet - This is huge. It lets you tether your droid to a computer and use the android device's data. (Warning: This is a bit of a grey area in terms of your service agreement. Don't use a boatload of data using PDANet if you do decide to use it)