If you want to do something totally free and absolutely amazing, take a walk across the Brooklyn Bridge. You walk on a special walkway above the traffic, which is speeding by underneath you, you have brilliant views of New York and it's all for free.
When you do it be sure to take the subway to Brooklyn and walk back to Manhattan - that way you get a great view in front of you all the time. Also, before you start the walk, take the time to grab a coffee and go down to the park on the waterfront in Brooklyn and just sit down and take in the view - awesome.
To walk from Brooklyn to Manhattan: Take the A train and get off at High Street. Then follow the signs, and the crowd, across Cadman Park to the Pathway. Once on the bridge, you'll have awesome views of Manhhattan.