I cruised with Carnival three years ago, and we're going again (with Princess) in a couple of weeks. Cruising is great fun!
I'd bring Dramamine with you if I were you. I brought some -- just in case -- and ended up needing it for a couple of days when the sea was a bit choppy. It's a bit rough on the stomach when you can't see the horizon (inside the ship).
As far as shore excursions -- if you can go snorkeling somewhere, that's a really fun activity, but I'm not sure how the kids would like it or if it they would be strong enough swimmers. Try to look for trips that are family-friendly. Book your excursions now if you can, or right away as soon as you get on the ship, because they don't have unlimited room. If you wait, you might not get to go on the excursions you want.
Remember that not everything on the ship is free. Carnival has a soft drink card that you can buy that is for unlimited soft drinks during the cruise. Certain restaurants and activities are not free. When we were disembarking from our first cruise, we walked by some people were trying to figure out how to pay for all the charges they had rung up on their trip before they could leave. Yikes!
What to bring... Hm... Lots of really good waterproof suntan lotion, for starters. Bug spray, I suppose, depending on what excursions you're doing (going into the jungle?). You won't need it on the ship by any means. Snacks to take with you on excursions (especially if you have kids)... Water bottles... A backpack to take ashore... Beach towels (there are "hotel towels" on the ship, but they are small...they drive me nuts)... Things for the kids to do in the room when you want to relax (DS, coloring books, toys, etc.)...
The kids' program on Carnival seems pretty nice. You can sign your kids up for it and they will get to do educational activities and will be watched when you want to have some adult time. I believe the kids' program is free.
