This Apple article may help.
http://support.apple.com/kb/HT1329
It's a bit long and seemingly complicated, but it is basically telling you to create the same
iTunes folder at the same location on both computers. So if you can get the files transferred over using some other method (such as over a network connection), it will work just as well. Note: You can use an external hard drive to move the music from one computer to the other instead of using the iPod.
You cannot use iTunes to go from iPod to computer (except for files purchased from the iTunes Store). The syncing using iTunes is from computer to iPod. However, there are third-party utilities that can transfer from iPod to computer.
Here's an article that may help, if you want to do it this way.
http://lifehacker.com/394046/copy-music-from-your-iphone-or-ipod-to-your-compute r-for-free
