How can I back up my app data?
If you take a full backup of your iPhone (right click on it in iTunes and choose Back Up) and then restore it to a new phone, all data will be preserved, including apps' data:


  iTunes will back up the following information
  
  
  Contacts* and Contact Favorites (regularly sync contacts to a computer or cloud service such as iCloud to back them up).
  App Store Application data including in-app purchases (except the Application itself, its tmp and Caches folder).
  Application settings, preferences, and data, including documents.
  


From iTunes: About iOS backups.