The Staten Island Ferry is free and gives a fairly good view of both the Statue of Liberty and the south side of Manhattan, if that sounds appealing at all. Many people do this just for the sights, so you can basically follow the waves of humanity as they depart and go right back on the next ferry.

The High Line is a very cool urban park built on top of an old elevated railway and is free to enter and exit at your leisure. The north end of it is right by Vessel, which is a cool bit of architecture.

The first floor of the Whitney Museum is free as well. And The Bronx Zoo is free on Wednesdays.