It is not possible to create Groups on the iPad. They have to be created with the service you are syncing with, and most services won't sync groups.
As far as I know you can only have/create groups if you sync with iCloud (iOS 5 only) or the old MobileMe service. You create the new groups via a desktop web browser at icloud.com for iCloud or me.com for MobileMe. Note, new MobileMe accounts are no longer possible.
If there are any other account types that can sync groups, I'd be happy to know of them. I have Yahoo, GMail, and iCloud. Neither Yahoo nor GMail (default or Exchange setup) can sync groups.