Oh wow, where to begin? First of all, congrats on getting the DROID. Many of my friends worked super-hard beta testing the software behind it, and so -- though I can't take any credit for this model -- I'm really proud of the phone and also excited about other Android 2.0 phones coming out in the future!
As for apps that I personally love... (and all are free, btw)
TRAVEL-ORIENTED:
- MyTracks for mapping where you walk.
- Tripit, which links to the
Tripit.com travel service online.
- Glympse, which lets you send a note to friends and *temporarily* enables them to track your location (great for when you're driving to meet friends)
- Google Latitude: A feature of Google Maps, but you have to turn it on. Super for sharing your location ongoing with a set of trusted friends (and seeing where they are, too).
- Google Sky Map: Just a bit of fun and fascination for checking out the stars and such when you travel, especially if you end up somewhere with a beautiful unobstructed night sky!
OTHER APPS I LOVE:
- Mother Ted: An app that connects you to videos from the inspirational TED conference. Just amazing to watch!
- Shazam: Listens to a song for you and tells you what it is! (artist, title) For use when you're in a bar or a passenger in a car listening to the radio, etc.
- 3banana: Use your Google account to sync notes/photos across computers/phones, etc. Very handy!
- Last.fm and Pandora: Outstanding radio apps! The Pandora mobile app convinced me to pay for a $36/year Pandora membership
Hope that gives you some good starting ideas :-)