There's a daily ferry service that leaves Nassau for Harbour Island and stops in Spanish Wells on the way. There's only one ferry a day in either direction - 08:00 from Nassau and 15:55 from Harbour Island. The trip takes about 2hrs 15mins and costs approx $110. This means you have about 6hrs on the island which is ideal to see everything and relax on Pink Sand Beach. The ferry terminal in Nassau is located near Bay St under the exit bridge to Paradise Island.
Up until recently, the only way to get to Dumore Town was to fly (from Nassau or Forida) to North Eleuthera, and then take a taxi to the public docks at the harbor, and then a water taxi to Dunmore Town. That is still a viable option (and not as hard as it sounds), but the new alternative is the Bahamas Fast Ferry from Nassau. They offer fast (2 hrs.) ferry service from Nassau to Harbour Island.
The whole island is 3 miles by about 1/2 mile wide. The main mode of transportation is by walking. Golf carts are also available for rent. This is a VERY laid-back place, and the only thing anybody does fast is talk.