My suggestion would be to get a bus pass and take as many of the routes as you can. Ask any of the locals what they enjoy, they'll fill your ear full. But, just riding around the island, getting off and walking, you can discover things you're interested in. It doesn't have to be what the "tourists" do. The locals are very friendly. Don't forget Horseshoe Beach.
Queen of waters, the summer months are when the cruise ships go to Bermuda. Late May, June, early July, late August, and September would be my suggestions. That is as long as the weather holds (no hurricanes).