If your backup is in iCloud, and your new iPhone is already set up, go to Settings>General>Reset, tap Erase All Content and Settings, then go throught the setup screens on the iPhone and when given the option, choose Restore from iCloud Backup and follow the prompts. (If it isn't set up yet, just choose this option when setting it up.)
Be sure your iPhone it connected to wifi and your charger while it is restoring the backup.
If you want to transfer the app data, text messages, your settings, etc. as well as your apps and other purchased media, you need to restore the backup of your old phone to your new one. There is no other way to do it. Syncing with iTunes will only sync your iTunes media to your phone, not your data.
Here is Apple's support article on this: http://support.apple.com/kb/ht2109.