Been to NYC 5 times - It's one of the best walking cities on all of the earth, so for the love of god, avoid tourist buses. Such a bad way to see a city like new york (or any city IMO). The subway can take you further distances.
Must Sees/Do's on a short time frame:
- Metropolitan Museum of Art
- Central Park
- Wandering around neighbourhoods: West/East Village, Chelsea, Little Italy, Chinatown, SOHO
- Eating lots of street food
- Explore architectural landmarks like Chrysler Tower, the UN, Prewar apartments of the upper east/west side, The Guggenheim, the Flatiron building, Rockefeller Center etc (best observation deck).
- Take a side trip out of Manhattan to Brooklyn (Prospect Park, Park Slope, Williamsburg etc) or Queens (Astoria) to see some more neighbourhoody places
Table 5 probably has some good advice.