Also, be sure the new computer is authorized on your iTunes account. On a Mac, in the iTunes menu bar, choose Store, and from the drop down menu choose Authorize This Computer. Windows should be similar.
If it doesn't do it automatically when iTunes is installed, you may have issues with your apps and any downloaded media from iTunes. Once the computer is authorized, you can plug the phone in and transfer the apps to the computer from the phone, so you don't have to download them again.
At least that is how it has worked for me.