To start with there are the beaches. From St Georges, Tobacco Bay, St Catherines Beach and Achilles Bay are all al ong walk or short mini bus ride away. Ft St Catherines might be fascinating as well. Shops and sights downtown include several places and events (like the noon "dunking" the kids might like. A nice glass bottom boat tour runs out of this port.
Hamilton has Ft Hamilton. Aquarium. Museum and botanical gardens. Horse drawn carraige rides. How about climbing to the top of the towr in the Cathedral?
Beaches easily accessable from here include Elbow and Horse Bays and all the beaches in between. Crystal caves is fun for kids and is accessable using the very cheap and friendly buses from Hamilton.
Dockyards area has shops, museums and old forts. Dolphin encounter run out of this area as well as a snorkel park.
"Excursions" include things like snorkelling and one extremely family friendly one is run by Jesse James Tours out of Hamilton. Unique for three stpos, including a small island , a wreck sight and a coral reef.
Look into the buses and ferries (and multi day passes ) to keep transportation costs down.
If you go to the Library and get either the Fromme or fodor book in the travel section on Bermuda,you can read up on the island and set your own excursions.You can do the whole island on public transportation.
We get their books for where ever we go and you can do it.After you read up If the book is due back befor you go ,take it back
and check it out befor you go and use as a guide.
We have been to Bermuda seven times and this is what we do.
You will not be the only people on the bus ,everyone from the ships use them.
I was born & raised in bermuda, so I may have some ideas.
Do your children like the sea & the beach? If they do I would recommend taking them to Horseshoe Bay. It's the biggest & best public beach in BDA. I don't know where your ship is docking in BDA, ( Hamilton, Dockyard or St. Georges) but you can reach Horseshoe, which is on the S. shore in Warwick parish by public bus (definitely cheaper than hiring a cab)
Also, you can take them sailing, fishing, & boating in general. "Glass Bottom" boat tour comes to mind.
We also have an interesting aquarium, which is located near Flatt's village in Smiths parish. Again, readily accessible by public transport (bus)
Also, there used to be a dolphin show years ago...Don't know if it still runs!
That should keep you & the Kids occupied for a few days! Enjoy the Island & the local hospitality & remember we drive on the left hand side of the road, and Bermudians are generally "Laid back", but quite polite & helpful.