The first computer you connect your device to is its "home" computer. The music library from that computer is copied to your device. To setup syncing with the new computer, you will need to connect the iOS device to your computer using the included USB cable. From there you can will need to select the syncing preferences. Please note, syncing with a new computer will wipe existing media from the device. I see from you post that all media has been transferred to the new computer so this shouldn't be an issue as you can sync the media back.