To the best of my knowledge it is not possible to create groups, or move contacts into groups in the Contacts app.
However, you can do both of these things at the icloud.com site and the changes will sync back to the iPad. You can not visit the icloud.com site via the iPad; you must use a modern computer browser. Firefox, IE, Chrome, Safari all work, provided they are reasonably up to dat.
If you do not use iCloud, then you can only manage groups via third party apps. While I know it is possible, I'm not familiar with any of the apps or their limitations.