ITunes help, please I'm desperate (had the problem 3 times throughout 2 years)?

Okay so apparently you can only have your apple product synced to one ITunes at a time. But I wiped my computer clean and re-downloaded ITunes and now my iPhone won't sync to ITunes. Is the there anything I can do to get my Itunes to copy the songs from my iphone and transfer them/save them onto the ITunes

Sign in to iTunes on the computer and authorize it on your home network.
Download the contents of your iTunes purchases from the store (under purchases)
You should be able to manage devices on the computer from your iTunes account.
Transfer purchases will add any non iTunes items back onto computer.