Stuff that EVERY Droid should have;
Handcent SMS - Puts Messenger, Google Talk, Google Voice messages all in one place with speech to text and pop ups.
Screebl - Saves battery and reduces frustration. Keeps your screen on while you're reading, turns it off when you're done automatically. It KNOWS when you're not reading.

Apps Organizer - Put one icon on your screen, but get quick access to your apps. Like folders but with customizable icons, and ability to close once you select one. For instance, I have a 'Games' button on my home screen that pops up all my games when I press it. Simple!
Barcode Scanner - No droid should be without this. Scan QR codes off websites like Androlib
A file explorer (Either ES or Astro) - Don't know why the Droid didn't come with an explorer. Either of these does the job well.
Stuff you'll probably want to get;
Pandora - If you've never tried it, then you need to right now. I have my phone playing Pandora through my bluetooth car stereo constantly. Never have hiccups and it sounds great.
A home replacement - Home++, Panda are the favorite free ones, GDE costs $3.25. All are very good. They allow you more screens, more options, and fantastic customizability. Which one you go with all depends on you, none is 'better' than the others.
SNESoid lite - How would you like to play every SNES game ever invented on your Droid? There's an app for that (but not on the iphone)!