Welcome to cruise-chat. While you will probably not heed this advice, I would strongly caution you about renting a moped to get around Bermuda. I visit Bermuda often and every time I have been there - I have seen more than one serious accident involving mopeds. Not only do they drive on the left side of the street (if you are not adept at this the roundabouts get quite confusing) but there are a number of blind alleys and hairpin turns that can be dangerous.
As for sights to see and things to do: what kind of things are you interested in? Forts and museums, lighthouses, fantastic beaches, snorkeling, nature preserves and botanical gardens, shopping, architecture???? There are a number of great restaurants on the island, but Lobster Pot in Hamilton is one of my favorite. White Horse Tavern in St. George's and the Hog Penny in Hamilton are also very good. If you like Asian Fusion than Cafe Gio in St. George's is good, and the Frog and Onion Pub at the Dockyard is a fun place with good food. Hope those suggestions help -