Ok, I figured out how to do it, here's the process:
First, this is how to transfer the ability to synch your iPhone from one computer to another without erasing your iPhone in the process. It will move over things such as apps and all the data in the apps (game progress, passwords, pictures, documents, etc). I use Pocket Informant for my calendar and it preserved all of that data. It also preserved my 3 stars on on levels status in Angry Birds. ;-) It will preserve each page and location of all the apps, folders, etc. It will also preserve settings such as wi-fi networks and email accounts. I believe it preserves everything that is preserved whenever you do a backup. I did loose custom ringtones, not sure why, but so far that's all I can tell I've lost. Not a big deal. Also, I was already synching contacts, bookmarks, and such over iCloud, so I didn't even try to get that stuff to synch. And I didn't try to get the iTunes music library to synch, or podcasts or movies. I was already moving my iTunes Library using other write-ups and I'd guess that is the best way to move it. FYI, the iTunes Library file is only music, it has nothing to do with apps and such, so moving that isn't enough.
I was going from a PC to an iMac and I have an iPhone 4s, but I think this would work for transferring the synch ability from any iTunes on any computer to another. Before you do this, backup everything you can in case something goes wrong. Be sure to do a backup (not just synch) to your current iTunes computer.
So here's the process, this is all done on the new iTunes you want to synch to:
1. In iTunes, get the latest software, authorize the new computer, and download all your previously purchased apps, music, etc.
2. Open the new iTunes, go into Preferences then Devices and select "Prevent iPhones, iPods, and iPads from synching automatically." You DO NOT want to synch when you first hookup your iPhone.
3. Now, hook up your iPhone. When it appears in iTunes under Devices right click and select Backup. You should also be able to backup with iCloud just the same, but I didn't do it that way. Once this finishes you can now backup and restore from this computer.
4. Now go through the menus at the top of iTunes and select all the stuff you want to synch, apps, photos, and such. But not stuff you synch over iCloud. You'll get a message that your iPhone will be erased and it will, click Ok or Proceed or whatever.
5. Now synch your iPhone. It might take a while and when this is done a bunch of stuff will have been erased.
6. Next go back to the left side of iTunes, right click on your iPhone and select "Restore from Backup..." This will take a while too, it will load everything right back onto your iPhone just the way it was before.
7. Last step, do another synch. This time, it won't erase everything as your iPhone is no longer new to this iTunes. Check everything thing over on your iPhone and it should all be back to normal. Now you are synching on the new computer.
If you ran into problems or if this didn't work, you should be able to go to your old iTunes and restore from backup to get your iPhone back to normal.