Today I'll show you a bit of the Lower Manhattan district, where few points of interest are situated (the Statue of Liberty and, of course, the World Trade Center Memorial), I must say we did this tour on our first day in NYC to have an overview of the city and the distance you are going to walk, it was chaotic and impressive in the same time.
First we went to the World Trade Center, since we hadn't bought any ticket before we just stayed outside. When you arrive you are in the financial district and there are several very very high skyscrapers as well as the new Freedom Tower (that was completed just after our journey).
I suggest to go on foot to Battery Park and have a walk, then use the ferry to Staten Island (if you don't like to walk the Metro station is South Ferry). This ferry is completely free and you'll have a cool view of Manhattan! If you are lucky - as we were - you'll get an old looking ferry with wood seats.
New York Street Food: since you are in the Financial District there are dozens of place where you can go, but on this website you'll get a list of cool cart where you can eat.