Apple provides a backup solution through iTunes and iCloud which will move all of your applications, photos, contacts, messages and application data to your new device without the hassle of reinstalling everything or trying to recover lost information. Apple and it's developers recommend performing a backup and restore via iTunes or iCloud to ensure data transfer. For more information, see: http://support.apple.com/kb/HT2109
Please note that if you tapped on "set up as a new phone" and/or downloaded the apps from the App Store, then that means you did NOT do follow the above procedure. Downloading an app from the App Store downloads a new, blank, copy of the app.
If you did not use the recommended procedure, then you may still be able to load the app and transfer your data. The App Store even has a "Purchased" section to let you easily find your old apps. In addition, we have a two backup features that will allow you to save your data out of our apps on your old device and restore it on your new device. Check out one of these FAQ's for an example of how to to do that: