The cheapest way to get to Nantucket island is to drive to Hyannis, a town located on Cape Cod in Massachusetts, and then take a ferry boat over to Nantucket. There are two companies that offer ferry boat services to Nantucket from Hyannis: the Steamship Authority and the Hy-Line. Each company offers two types of boats - a slow ferry that takes over two hours, but costs less, a "fast ferry" (hydrofoil) that takes about an hour and is more expensive. Cars can only be taken onto the slow ferries, and require reservations in advance.
For the fit there are bicyle rentals.....mopeds for the less so. I forget my credit card so I had no choice but to use public transportation. It actually turned out to be the best.
You can get a daily unlimited pass for $5 and visit the surrounding towns quickly.