Disable "auto-sync" in iTunes first. Preferences - Devices - Prevent iPhone/iPod from automatic syncing.
To be on the safe side, import all of the photos/videos (Camera Roll) from your current iPhone to your computer.
Backup your contacts on a compatible programme on your computer (Outlook, Address Book, Windows Contacts etc) or in iCloud.
Now follow the steps mentioned in this article.
Later, sync the contacts with your new iPhone from the programme/iCloud you used to back them up.
You may transfer the data from your iPhone4 to iPhone4S by using iTunes or iCloud. If you choose to use iCloud, you may:
Back up the old iPhone manually with iCloud by opening “Settings, tap “iCloud”, navigate to bottom and tap on “Storage & Backup”, then tap “Back Up Now”
Boot the iPhone 5 and walk through the easy setup, choose either “Restore from iCloud Backup”
Wait until the new iPhone restores from your old iPhone backup, this can take a while depending on the size of your backup, how much stuff you have on the iPhones, and the speed of your internet connection
When finished, your new iPhone will have everything from the old iPhone and you’re ready to go!
To transfer data from one iPhone to another:
1. Connect two Devices to PC.
2. Choose the category list of one device that you need to export files
3. Check the files that need to export in the Content Panel and click "Transfer file(s) from device to device" button
4. All the checked files from one device can be transferred to the other one.