How to move an iPod or iOS device from one computer to another without losing anything?
Your question can be rewritten in short form:


iTunes: How to move your music to a new computer
Where are my iOS backups stored? - expand the section About Syncing and Backups and you'll see


iTunes places the backup files in the following places:


Mac: ~/Library/Application Support/MobileSync/Backup/
Windows XP: \Documents and Settings(username)\Application Data\Apple Computer\MobileSync\Backup\
Windows Vista and Windows 7: \Users(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\


I linked to the official articles since things do change and cutting and pasting the steps today might not be right if things change in a month or two. Don't worry - the article about music moves everything except the iOS device backups. If your device is working, there's no need to move backups unless you want to keep older ones or save time the first time you sync with the new iTunes.