How do I backup my iPhone and restore it on another machine?

"Install iTunes on the other machine and sync your iPhone with that machine. iTunes will automatically backup your device so you can restore it whenever it is needed.

iTunes will automatically backup your iPhone when you sync it, or when you update or restore it.

You may have to move your whole iTunes library to your new computer. You could make a backup of your library by going File > Library > Backup to disk and then load the resulting disks into the new computer. Alternatively, you could try manually moving your library."

"Whenever you sync your iPhone to a new PC/Mac, you are able to sync it's contacts, calendar, call history to the new computer. Only the videos, photos, and music can't be sync FROM the iPhone.

If the question you ask was about how to sync the essential content (i.e. contacts etc), you have no need to worry since iTunes will merge the contacts from your iPhone with the contacts in your new PC/Mac.

Of course later on you'll have to migrate your iTunes library to your new computer afterwards"