Syncing to a new iTunes library or computer will erase your phone. Only if you back up manually before syncing, you can restore your device from that backup again. A manual backup does not include the sync process.
Disable autosync in iTunes, connect your phone to your new computer and right click on it in the device list and choose backup. iTunes will backup your device without syncing.
Transfer your purchases the same way, choosing "transfer purchases" this time.
When you connect your phone for the first time, all media content will be erased. But you can restore your settings and app data from your manual backup afterwards.
Don't forget to set up at least one contact and event on your new computer to be able to merge calendars and contacts when you sync the iPhone for the first time.
Music is one way only, from the computer to your device, unless you bought the songs in itunes and transferred your purchases.
There is 3rd party software out there, but not supported by Apple, see this thread:http://discussions.apple.com/thread.jspa?threadID=2013615&tstart=0
About backups and what's saved:iTunes: About iOS backups
How to back up and restore:http://support.apple.com/kb/HT1414
How to download apps for free again:http://support.apple.com/kb/HT2519
Saving other data is also described here. How to back up your data and set up as a new device
I have been an Iphone/Mac/Apple person for well...ever. But when it came to syncing and backups, I handed all of that over to my ex. I have been on my own for over a year now and have been searching for a way to sync. I have been so incredibly confused looking up help topics and trying on my own, I have lost a ton of data and somehow created duplicates of all my contacts!
I haven't gone to the links yet, but at least you have divided them to say this link does this and this link does that! Thank you so much for taking the time to explain this process to the above question! In doing so, you also helped me today! I really appreciate it!