Great answer from amandaafoote. I’d just add that if you turn iTunes’ “Keep iTunes Music folder organized” option on, when you add the folder to iTunes, it will rename all of the MP3s and folders from F00 etc to something human-readable and useful. (That option is under Advanced > General).
Or, if you happen to be on Windows, you can copy the folder the way she said, but then use a program like Tag&Rename to fix the file names for you.
Also, one program that’s done a good job of getting music off my/friends’ iPods for me in the past, without having to use iTunes at all, is YamiPod. It’s free; it runs on Mac, Windows and Linux; you can put it on your iPod and run it directly from there on any computer (no need to install it first); you can play your music using it, right off the iPod; and it’ll let you copy just a part of your library off the iPod, including renaming the files for you.