Syncing an iPhone or iPad to a new computer using iTunes may cause some troubles with loosing data (music, photo, movies,...) or having to download all apps.
At for just apps, it does not matter since you can re-downloaded all paid apps using your iCloud account. However, if your current devices contain so much data, look at the following tips to Sync iPhone or iPad to a New Computer Without Losing Data.
Sync iPhone or iPad to a New Mac Without Losing Data.
Quit completely iTunes on both Mac PCs and make sure the iPhone or iPad is disconnected from the both Macs.
Access the Home folder and copy and move the iTunes directory from the old mac to the new mac.
Copy and move the backups from the old Mac to the New one.
Copy com.apple.iTunes.plist from old Mac, then paste to replace into the new Mac.
You now can connect your devices (iPhone or iPad) to the new Mac, then sync without loosing Data.
Syncing an iPhone or iPad with a New Windows PC without loosing Data
Basically, the instruction is the same as when you deal with a Mac, but the files and folders needed to be moved from your old PC to the new one are in different locations.
The USERNAME is your user account name. The iOS backups is saved in the following direction:
C:\Documents and Settings\USER\Application Data\Apple Computer\MobileSync\Backup