I don't have the droid yet, but ill have it within the month, so im doing research on the apps i'm gonna get.
First and foremost, i'm gonna get the dolphin browser. Tabbed browsing with multitouch pinch zoom? count me in =]
I don't use itunes, but if you do, you can get
doubletwist which is a pc-side program that helps with managing music for android OS's
The GoogleVoice app seems pretty neat, and ive heard people get free texting from it.
My most anticipated app would be ringdroid, which lets you make awesome ringtones from music right from your SD card, without having to use your computer or anything. (im a heavy ringtone maker, so this is a must have app for me)
If you're paranoid about your phone being lost, look into the mobile defense app
I know there area few more apps i was looking into, but i stupidly didn't bookmark them. Most of the apps I mentioned came from
this link, which i found in a different thread. Along with neat apps, it also has a lot of information about the phone itself, and ways to customize and optimize your new droid