you have the obvious sites of Empire State Building, Chrysler Building, Central Park and Times Square. i Say focus on these sites. St Patricks catherdral. Madison Square garden, Yankee Stadium, Statue of Liberty and Ellis Island (even though they belong to New jersey) Rockefeller Center, Lincoln Center, Little Italy, Brooklyn Bridge the list goes on...
take a subway
Ground Zero
Statue of Liberty
Radio City Music Hall
Empire State Building
NY Stock Exchange
Yankee Stadium
Museum of Modern Art
Museum of Natural History
Museum of New York City
see a Broadway play with half price tickets the day of the play
find an ethnic restaurant you have never tried before
So allow me to list great places..
Lower Manhattan: World Trade Center site; boat to Ellis Island and Statue of Liberty (same boat, one price, get off and explore, take later boat onward, start early as LONG lines form); Fraunces Tavern; Chinatown (eat at Wo Hop on Mott Street); South Street Seaport; Chase Bank Money Museum.
Midtown: Empire State Building; theater district (great restaurants just uptown of theaters); window shopping along Fifth Avenue 47th to 58 Streets; United Nations (the gift shop is a great place for souvenirs); FAO Schwartz (the apotheosis of toy stores, and includes an ice cream parlor).
Uptown: Central Park and zoo; Metropolitan Museum of Art; Museum of Natural History/Rose Space Center.
Way Uptown: The Cloisters (most beautiful spot in NYC, has the unicorn tapestries and other pre-Renaissance art, out door gardens, and music); Dyckman House (British army HQ in the Revolution).
Bronx: Zoo; botanical gardens
Brooklyn: Aquarium; Coney Island
Queens: Bowne House (stop on the Underground Railroad, good restaurants nearby); Flushing Hall of Science and the adjoining zoo with an aviary where you walk at tree top level to see the birds).
Staten Island: view the harbor from the ferry, then on the SI side take a bus to Richmondtown Restoration (restored colonial village, free admission); Marchais Tibetan Museum (only Tibetan Museum in the entire USA); Conference House, site of the only peace conference between American (Ben Franklin & John Adams) and the British during the Revolution. While on SI eat at the Bay Street Bistro, really delicious Mongolian food!
If you want the exact locations of some places to eat,let me list it over here
The choices are so great that it is almost overwhelming. I've lived here so long and there are still restaurants that I've intended to get to and just haven't had the time.
Here are a couple of suggestions that will give you a feel for some of the types of places here.
*Big Nicks is a must. It is a real NYC joint. An Upper West Side institution. Tiny space but a huge menu and amazing food. Cheap prices.
*Philip Marie in the West Village. The Brunch is amazing.
*Mary Ann's in the East Village for Mexican.
*Balduccis for Indian Cuisine (several locations, my favorite on 6th Ave in the Village)
*Angels Restorante - Upper East side for Italian at 1135 1st Ave, New York, NY
4.1 mi NE - (212) 980-3131
*Little Poland - Great Polish food.
Outside of Manhattan try 5th Avenue in Park Slope or Smith Street in Cobble Hill, both in Brooklyn.
*Desert oh yea, Juniors for Cheese Cake or Veniero's for everything else.
get a copy of the Village Voice - free on the street
get a good guide book - there is no end of things you can do in New York....