The dockyard is quite distant from the rest of the island. By far, the best way to travel from the dockyard to Hamilton, and especially St. George's, is by Ferry. A taxi ride to Hamilton will cost almost $50 with tip and takes about 30 minutes. Buy a 2 or 3 day transportation pass and get unlimited ferry and bus service. There are special buses taking people directly from the ship to Horseshoe Beach, which saves time and money.
Don't book shore excursions through the cruise line. Instead, visit the tour booth at the dockyard. Taxis are a great way to see the island and cost $40 an hour for sightseeing (1-4 people)- much better than being crowded on a tour bus (which can't go to some parts of the island due to size limitations).
Few ships visit St. George's anymore (Veendaam currently does). Evidently, the Bermudian government wants the cruise lines to use their ferries to tender passengers into St. George's and the cruise lines want to use their lifeboats. So we get an extra day at sea instead of another day in Bermuda because the two parties can't agree. Disappointing- I'd rather be in Bermuda than at sea!