To get a great experience of NY, jump on the Staten Island Ferry. It gives you a great view of the downtown skyline, Brooklyn and the Statue of Liberty. Also, it's free, runs every few minutes, and real New Yorkers will be alongside you. Once you get to Staten Island, get right back on a return ferry and enjoy the fab experience again.
To get a similar view, you could take the Statue of Liberty Ferry, which is not cheap and usually has a long queue. Only take that if you want to tour the Statue of Liberty, which is usually full of kids and queues.
Battery Park, which is very downtown; the end of Manhattan. The park and ferry terminal are very well sign posted.
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