You can only sync to one computer. You would have to backup your iTunes library and transfer to the new one. The iPhone is not a storage device.
At this point, if you don't have access to a backup, you would have to transfer purchases
For non-purchased content you will need a 3rd party program to transfer off the phone.
The iphone is not a storage/backup device.
It simply mirrors the selected content of your computer. If you sync to another computer, it will mirror the content of that computer.
It has always been very basic to always maintain a backup copy of your computer for just such an occasion.
Use your backup copy of your computer to put everything on the new one.