If it's a pink sand beach you are in search of -don't waste your time at King's Wharf. The majority of the Bermuda beaches of your dreams are along the south shore. Both beaches you are considering are spectacular, although I prefer Elbow Beach only because it offers a longer stretch of sand for the beach walks I so enjoy. (Warwick Long Bay is also great.) You can take one of the local buses to either beach, but in the interest of time I might consider taking a taxi (be prepared they are expensive). Or you might want to consider taking the local bus (there is a terminal with information at the dock at King's Wharf)to the beach in the morning and then paying the extra fare for a taxi back to port to maximize your beach time. From the Dockyard, Horseshoe Bay is the closest.
Hope that helps!