The below way tells you how to sync iPhone to new computer without losing data, hope it can help you a lot.
To Sync iPhone With New Mac
1- Quit iTunes on both Macs
2- Disconnect your iPhone from both Macs
3- Open the Home folder
4- Copy iTunes directory from the old computer to the new one
Located at:
~/Music/iTunes
5- Open user library directory and copy the backups from the old computer to the new one
Located at:
~/Library/Application Support/MobileSync/
6- Also copy over iTunes Preferences files
Located at:
~/Library/Preferences/com.apple.iTunes.plist
7- Launch iTunes in the new Mac up
8- Authorize your Mac to Apple Server
9- Connect your iPhone and sync as usual
To Sync iPhone With New Windows PC
1- Quit iTunes and make sure all Apple's services are not running
2- Launch Windows Explorer / My Computer up
3- Copy the following directories and their contents
On Windows 7 (Replace "USERNAME" with the users account name):
C:\Users\USERNAME\Music\iTunes
C:\Users\USERNAME\AppData\Roaming\Apple Computer\MobileSync\
C:\Users\USERNAME\AppData\Roaming\Apple Computer\Preferences\
On Windows XP:
C:\Documents and Settings\USER\Application Data\Apple Computer\MobileSync\Backup
4- Now paste / put those folders to a new computer within the same path
5- Once done. Launch iTunes in the new computer
6- Authorize the new computer to Apple Server
7- Connect your iPhone and Sync as usual.
That's it. Do not forget that there is limitation of the number of your computer that can be authorized.