Greetings Your Highness!
There are a lot of activities onboard and it will depend a lot on your family, kids ages, etc. But generally speaking...
- Formal Night - a lot of folks on Disney cruises don't like to dress up and I roll my eyes a bit at it too, but then you see the folks who go all out and you think to yourself, "gee, I wish I'd taken this more seriously..." It's awesome to see a couple in formal attire when one of them is in their dress uniform from an armed service. Or even just spiffed out in a tux or ball gown, etc.
- The shows - seems obvious, but there are folks who just don't go to the shows. We make it a point to attend all of the shows as a family and we generally enjoy all of them. Especially the variety acts!
- Trivia - granted this caters to a certain type of person, but it's a ton of fun and you can win some cool prizes too.
- Anyone Can Cook - the chefs will show you how to prepare a dish that's served onboard and then give the audience samples of it and a recipe for how to make it.
- Ship Tours - Disney doesn't do a "behind the scenes" tour like other lines, but that doesn't mean there isn't some neat stuff to see! If you're lucky and can catch the back stage tour of the Walt Disney Theatre, that's super cool!
- Ice Carving - this is usually on deck 9 at the FunnelVision stage. The kitchen staff will carve a huge block of ice into something cool (pardon the pun)...
- Midnight Buffet - after Pirates In the Caribbean night, there's usually a midnight buffet in Cabanas on deck 9 aft. Check it out to get crepes, turkey legs, and some other awesome food you don't get on the rest of the cruise. Note that this isn't on all cruises.
- Officer Pin Trading - if you're into pin trading, there are a couple of times you can meet the ship's officers and trade pins with them. This is usually in the lounge outside of the two stores on deck 4 midships.
Have a blast and I hope this helps!