If you've been using an up-to-date version of Dropbox, it's likely the service has also been backing up your photos automatically whenever you connect a USB camera, iOS device, or memory card (provided that you haven't turned down its request to do this). Download a copy of Dropbox's mobile app to your iPhone, iPad, or other mobile device and check.
Your iTunes purchases
If most of your music was purchased through iTunes, it's easy to restore your library. First, make sure that you're signed into iTunes with the account you used to purchase your media. Now click on the iTunes Store button in the top-right of the iTunes window, and click the Purchased link that appears below on the right side. In the resulting screen, click the Music tab and then the All Songs entry. All the music you've purchased will appear in a list. Click the Download All button to download all your past music purchases. You can also redownload purchased movies, TV shows, TV seasons, and books using this same technique.
If you subscribe to the $25 per year iTunes Match service, then you can download any music that you've either matched or uploaded.
Anything on your iPhone
You might be carrying copies of important data right in the palm of your hand. At the very least, your iPhone probably contains a chunk of your photo and music library.
Your photos: Your photos will download automatically when you connect your iPhone to the new computer.
Ripped music: If you spent hours ripping CDs to your iTunes library and the songs remain on your phone, there's hope. Apple doesn't make it easy to restore music from your iPhone to your Mac, but it is possible. With a tool such as Ecamm Network's $30 PhoneView, you can extract data and media from your iOS device.
Just jack in your device, fire up PhoneView, select the kind of media you want to extract, and--if it's music, for example--copy it directly to your iTunes library by clicking Send To iTunes.