I lived in Bermuda for 3 years -- you will have a great time.
First, I woudn't discourage you from renting a moped. It's the best way to see the island. They are safe if you are a safe driver. The Bermudians are very respectful of their visitors and I've never seen an incident of them playing around with tourists. And, I wouldn't characterize the bus drivers as maniacal. Follow the rules of the road, drive the speed limit and you'll be fine.
And wear sunblock. You'll have a better time if you aren't burned.
Horseshoe Bay Beach is the most beautiful on the island. It's on the South Shore and there are dozens of beaches you can stop by to tour. Elbow Beach is one of them. Also on the South Shore is Ariel Sands Hotel. Great place to stop for lunch and it's owned by Michael Douglas and his mother's side of the family. His mother is Bermudian. The lobby is decked with photos of Michael on the sets of all his movies, his extended family and CZeta Jones of course. If you're lucky, you may run into them because they live on the island and frequent the restaurant.
Downtown Hamilton has great shopping, especially Front Street. Triminghams is the "Filene's/Nordstroms" of the island. Coopers, the English Sports Shop and the Irish Linen Shop are great too. If you like rum, stop at Goslings to buy a bottle. "Dark and Stormy" is the island's unofficial drink. (OK but don't drink alot of them and then drive the mopeds!!!)
There's a birdcage in the center of town where the policeman (Bobby) directs traffic. Good photo op.
I love love love the Botanical Gardens. It's worth the trip if you love flowers and gorgeous settings.
The aquarium and crystal caves are good for the kids. (You didn't say how old they are) Also, I'll bet they will like the Gibbs Hill Lighthouse in Southampton. Excellent views of the entire island.
Have fun!!! Bermuda is immaculate, gorgeous and the Bermudians will make you feel welcome. Good luck for good weather!