New York, Manhattan, there are various city tours based on popular tv-shows, pick one you love to watch and see all the famous places where it has been filmed. Go see a show on Broadway, off-Broadway, or even off-off-Broadway. You can get cheaper tickets at the ticket sales place (name and address? TKTS). They sell tickets for that day, so whatever is still available. It may not be for one of the very popular shows of that moment, but still very good ones and at a reduced rate. Take the free ferry to Staten island, and get back on to go back to Manhattan immediately, nothing to see at Staten. But it’s a free trip in the big lagoon of New York, great to see the skylines and the Statue of Liberty. Walk around Central Park, there are lots of little trails around. Go to the MoMa Museum (lots of well known art, free entrance on Fridays and live music), Guggenheim different exhibits, the building is also interesting work of art itself (just like in Bilbao, Spain). Walk the Brooklyn Bridge. Go to Ground Zero and the 9/11-museum, very impressive.
Tour U.N. Building, interesting facts. Bring passport. Empire State Building around sunset, great views of the city during sundown.
Get a Metro Card, it is so easy to get around in the city with the subway.