I currently have a phpBB forum and two servers, one of which I'm going to be moving to momentarily. Just to make things simpler on my part, is it suggested that I move my site to the new server first, upload the phpBB3 database (it's small, just a couple of MB at the moment), and THEN install/import xenForo?
Also, if that's the order I should handle things, do I need anything besides my old database? I don't need to upload any files from phpBB3, do I?
You need the attachments and avatars from the file system.
While you can do a server move + import in one motion, I only recommend combining the two if you are practiced with both. Things can go wrong with a move as well as with an import. It is sometimes best to move first, let it settle in, make sure everything is working, then do an import.