Have you decided where to go camping yet? I would recommend Malaekahan Campgrounds on the windward side of the island. A bit more out in the country, so it'll just be you, your family, and the beach. Can't beat that! Though, this campground requires reservations so I'd put in a request
HERE. To see if your vacation dates will work.
If that doesn't work, try out Ahupua'a'O Kahana State Park. It is also on the windward side. There's hiking trails in addition to the beach.
Personally, when I was young, my family camped at Bellows Field Beach Park. There's a beach with a lifeguard and plenty of room to use your snorkeling, and body-boarding gear.
As for activities with little to do with the city....what did you have in mind? Somewhere outside of Waikiki or Honolulu. Or something like a local festival?