How do I un-sync the first computer my iPhone synced with and ensure another computer is able to sync the apps to my phone?

Copy the entire iTunes folder (not just the music) from your old computer to your new one using one of the methods discussed here: http://support.apple.com/kb/HT4527. If you synced any photos from your computer to your phone. If you are syncing contacts and calendars with a supported program on your computer, you'll also need to transfer these to your new computer.

Apps are in iTunes Music>Mobile Applications. If you are you getting an error message when you attempt to sync that iTunes has detected that your device was previously synced with another computer, then copying the entire folder, which contains library files that associate your phone with the iTunes library, to your new computer will solve this. You will no longer get this error message when you attempt to sync.

You can't, it would replace it. I was under the impression that they were both yours, and that you were trying to migrate from an old on to a new one. If the content on the two computers is different, and you are trying to re-establish syncing with this computer, then follow this guide: Syncing to a "New" Computer or replacing a "crashed" Hard Drive. If you are using iTunes 11, go to View>Show Sidebar in order to follow this guide. Also, Transfer Purchases (step 5) is now in the File>Devices menu.