For first time visitors to New York, following some walking routes can be a good way to explore the city. The best way to explore New York is on foot. The grilled shaped Manhattan streets with vertical avenues and parallel streets make walking easy from one block to another. The subway, though it may look old and dirty, is also easy to use and a week metro pass for thirty dollars will let you to move fast in the long island of Manhattan.
First the Midtown: start with the must visits to landmark buildings such as Empire State, Grand Central Terminal, Flatiron Building; then head to south Manhattan to visit the 9/11 memorial and climb to One World observatory to watch Manhattan skyline; cross the Brooklyn Bridge on foot or by ferry. Take the Circle One ferry on Hudson River from Northwest pier for a two hour ride around the island of Manhattan.
You may spare at least your three days to the museums: The MET with three museums, the main one with its collection of antique civilisations at the Upper East by the Central Park, The Met Breuer for contemporary art exhibitions couple of blocks away on the Madison Avenue; The Moma to see the legendary avant-garde art works from the 30s; the Guggenheim to see any exhibition at this spiral shaped building. Do not forget to check the art galleries in Chelsea. Central Park and gospel in Harlem will be a nice Sunday morning activity, whatever religion you may believe listening to gospel will give you goose bumps.
For night life there are several options: the musicals in Broadway, the jazz bars in Greenwich, comedy clubs, the classical music concerts in Lincoln Centre, home of New York Philharmonic Orchestra. The Museum of Natural History is another place where jazz concerts take place.