Apple - 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.

Vlad's answer is good, and he's absolutely right.

An alternative to iTunes would be to use an iOS device browser. This would be especially good if you only wanted to back up one app's data rather than the whole device. Options of these include Phone Disk and PhoneView.

To backup your device to iCloud:

1. Launch the Settings app.
2. Go to iCloud > Backup & Storage.
3. If you enable iCloud Backup all your data will automatically be backed up each night.
4. Or you can press Back Up Now to immediately perform a backup of all your data.

To restore your data to the same phone:

1. Make sure you backed up and that the backup completed successfully.
2. Delete all data on the phone by going to Settings > General > Reset > Erase All Content and Settings. -- WARNING: This will delete all your data, so make sure you have it backed up, especially anything not syncing to iCloud.
3. Follow the steps below for a new phone.

To restore your data to a new Phone:

1. During the Setup Assistant, choose to Restore from iCloud Backup.

Notes:

• This will only work if you have less than 5 GB of data on the device. Otherwise you will need to buy more space from Apple or choose to not sync certain things (e.g. Photos and Videos). If do the latter, realize that when you erase all your settings all that data will be gone unless you copied it somewhere else besides iCloud.