How do I extract the data for a single app to transfer to another device running the same app? I want to move a personal wiki from an iPad 1 to an iPad 3.
You can copy any file from and to your ipad manually using a software tool like iFunBox without the need for jailbreaking. I use iFunBox to backup save-games and app settings.
The exact answer depends somewhat on the app in question, however there are several apps available which will make your iOS device ‘browse-able’ from your Mac or PC.
iExplorer (Mac or PC) http://www.macroplant.com/iexplorer/
PhoneView (Mac only) http://www.ecamm.com/mac/phoneview/
iFunBox (Windows only) http://i-funbox.com