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Block Island has a lot of places to explore: lighthouses, beaches, bird sanctuaries, and even a salt pond. Now, word to the wise, the area south of the ferry (towards Southeast Lighthouse) is a bit hilly, so if you are not an experienced biker, you may want to stay clear. There's a nice hike that way, some of which you can do along the beach (as opposed to walking along Spring Street).
The area to the north, along Corn Neck Road, is flatter and easier for bikes to navigate, except for the section near North Light House, which has a bit of a dip. Just be aware that either roads can have quite a bit of vehicular traffic and there are no bike lanes. For a daytrip, if you aren't comfortable riding with traffic or have young children, you may want to leave your bikes on the mainland and just stay near the ferry area beaches. Or take a taxi to the farther reaches.
If you have kids, the closest beach is Kids' Beach, just north of the ferry. It doesn't have facilities, but the Visitor's Center does (as well as, lockers). Fred Benson Town Beach (the next beach north of Kids' Beach) is the only beach with full facilities - changing rooms, flush toilets, food and drink. Showers cost $2. There are no free outdoor showers, although there is a foot washing faucet to the right of the concession stand.
Hike along the beaches - barefoot, if possible. It's a great way to go!