One of the best and most inexpensive activities on the island is to pick a harbor side beach, bring a chair and a cooler and watch some of the biggest yachts cruise on by.
You can take the Nantucket shuttle service, taxi, moped or bike, but if you can hold out til 1230pm and wander to the visitors center in town, a free shuttle awaits to whisk you away to an oasis several miles from town. It is an actual brewery that has turned their grounds into a daily party. It consists of little buildings for all the alcohol they produce. They also have live music, food trucks and corn hole.