If you haven't synced you phone with iTunes before the first thing iTunes will do is delete all iTunes media (apps, music, podcasts, etc.) from the phone, then replace it with the media in the iTunes library you are syncing with. Establishing syncing for the first time without losing your data has to be done carefully. This guide will give you the process for doing so: https://discussions.apple.com/docs/DOC-3141. There is a related guide here as well: https://discussions.apple.com/docs/DOC-3991. However, as covered in these guides, only music purchased from iTunes can be transferred from your phone to your iTunes library. Transferring music from other sources from your phone requires 3rd party software such as Touch Copy.
If all your music is purchased from iTunes, it can be download directly to your phone for free without syncing (see http://support.apple.com/kb/HT2519). Another option to get your music without syncing is to subscribe to iTunes Match (see http://www.apple.com/itunes/itunes-match/). This will scan your iTunes library, match songs already in the iTunes store and upload any songs in your library not in the iTunes store, then make your entire music library (up to 25,000 songs) available to download from iCloud directly to your phone. (With iOS 6 due out next week on the 19th, iTunes Match songs can also be streamed to your phone without downloading and taking up space on your phone.)
You can manually move some of your files and information to a new Mac or PC, but the easiest way to start using your iPhone or iPad with a new computer is to get an app like iPhone Transfer.
Be sure that you use one of these tools and follow the steps outlined below how to sync iPhone files to your new computer.
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<Edited by Host>
Sometimes people make things so complicated.
A. Copy your iTunes folder from your old computer to the desktop of the new computer.
B. Rename the iTunes folder on your new computer to something like "EmptyITunes". Its usually
resident in "yourAccount/Music".
C. Move the "EmptyITunes folder to the desktop of the new computer
D. Move the iTunes folder on the desktop (from the old computer) to the same place
where the "EmptyITunes" folder was.
E. Launch iTunes. You're done.
F. Plugin your iPhone, it will sync as if its the old computer, since of course, the files are the same.
Once you are comfortable, deauthorize the old computer and delete the "EmptyITunes" Folder
Always backup every hour with Time Machine.