There is nothing to do in St. George's other than take a bus to Hamilton or take a ferry to the Dockyards. After a 30 minute ferry ride from the ship to St. George's, we walked the entire length of the town in 15 minutes and then scrambled to figure out what else to do with the day. Use St. George's for an excursion.
Hamilton was a cute little island town. The Veendam docks right in downtown Hamilton, with easy access to ferries and buses to the beaches. Do not get a taxi - take the bus. The 10 & 11 go from St. George to Hamilton, the 7 takes you to the pink beaches - Horseshoe, Elbow, etc, and the 1 takes you to Tucker Town. The ferry to the Dockyards is literally a 2 minute walk from the ship. Hamilton is a nice hub to start from, but inevitably you'll go outside for beaches, golf, etc.