Mook,
I switched computers not too long ago. I used a program called CopyTrans. It cost me 15 bucks, but well worth it considering even downloading the 3000+ songs illegally would take HOURS.
It took like half an hour to back up the songs to your new computer, then you sync. Really easy. Just don't do what I did and waste time using the free one. They don't tell you it but it only backs up 200 or 300 songs or something, then everything after that you need the paid version for.