Honestly, when it comes to cruising, I think 20 yr olds are in a bit of a "lost area". There are no activities clearly aimed at your age group. You can go to the clubs onboard, but can't drink.
You are likely to find some others in the same situation as you, and have some people to hang with. But.. probably not a huge group.
You could get your own cabin onboard since your father will be onboard, but it would be costly to pay single supplements for both of you.
A cruise to Europe, like the RCI cruise you mentioned, might work very well because those itineraries are very port intensive, with lots of incredible sights to see. You wouldn't really have time to worry about all your other concerns; you'd be having to much fun together seeing the ports.