c:\~user\AppData\Roaming\AppleComputer\MobileSync\Backup
Not sure of the file name, as I'm not on the one i use with my phone. You should, however be able to create a backup of your phone on the new computer when connecting it to iTunes. Make sure automatic syncing is disabled in iTunes, connect the phone, and right click on the phone in the device pane and select backup.
You'll need to copy your library over to the new computer as well, and authorize it through iTunes. If you don't copy the library, if you try and sync the phone, you will lose the library and all the music on your phone.