How long will you be in town?
Eight weeks?
Twelve weeks?
EXCELLENT!
See everything suggested above, (Museum of Modern Art, The Metropolitan Museum of Art, The American Museum of Natural History, the Guggenheim Museum, a Broadway show, Central Park, Empire State Building, New Freedom Tower, Yankee Stadium, the NY Public Library on 5th ave, the Plaza Hotel, stroll down FIFTH ave and marvel, the statue of liberty)
add the Hayden Planetarium and the United Nations Building. Then head to the northern tip of Manhattan and explore The Cloisters Museum. Head over to Staten Island and investigate the long-gone site of the first Bethel Farm/Bordello, then find the Church upon whose grounds is a vast EMPTY space where old-time Dubs like W.E.vanAmburg are buried, all sans-headstones. I can supply directions if you wish.
See the magnificent Verrazano–Narrows Bridge, see Coney Island. If time permits, drive all the way out to Montauk Point (116 miles):
You will see "the island" magically transform from a busy metropolis to something more like cape cod; that eastern tip is warmed by the gulf stream and you will find remarkable life in the tide pools out there.
You can find the site of Tesla's energy transmission experiments at Wardenclyffe Tower (1901–1917), also known as the Tesla Tower, an early wireless transmission station designed and built by Nikola Tesla in Shoreham, New York in 1901- 1902.
It is on the North Shore of Long Island: