Sounds like your Droid has a custom rom flashed. It is not Nexus 1. they are completely different phones. It is likely the 2.1 operating system. Checking in "about" will confirm. If so, going with the 2.1 OTA update is actually a step back. Go into Astro, look through your SD card, and delete (long press, edit, delete) the file "update". It will quit bugging you to update.
As far as recommended apps, figure out what you want it to do. You'll get better personalized recommendations. Have you gone through setup for your own gmail, or is it still on the previous owners? Definitely don't wanna waste money on paid apps until it's on your gmail acct. Don't want previous owner getting email confirmations with your credit card info either.
Some recommended apps once you know where you're at (rom version, gmail acct, etc) would be Rom Manager paid version. This gives you access to reputable roms, themes, fonts, and kernels in a noob-friendly UI.
Next recommendation would be
Swype for Droid. Takes a few days to get used to, but you will dig it once you're used to it. When you click install, it will take you to a setting to enable installs from outside Market. Enable that, then click install again.
For music, search market for "fabrantes" and download the only app that shows up, which is called 3. There is also Astro Music Player Beta, which has a functioning equalizer, but that ones a little buggy still.
Do some reading on flashing roms. you will be in much better shape when you understand what you are doing.