New York is more of an atmosphere than a specific sight -
You can start by sitting for a few moments by the pond on Central Park south ( featured in some movies), off Fifth Avenue. Walk south on Fifth Avenue - you can see the Grand Army Plaza and the famous Plaza Hotel ( I'm sure you have seen it in some movies, too) and marvel at this beautiful avenue and some of the famous retail buildings lining it. There's Bergdorf Goodman ( temple of retail therapy - you don't have to buy but you can ogle), Tiffany, Trump, etc...until you reach the most famous church in New York, the gothic-looking St. Patrick's Cathedral and right beside it is now the landmark building of Saks Fifth...If you cross the street, there is the Rockefeller Center and its famous plaza ( best time to be here is at Christmastime)... you can also see famous television studios ( if you are very early, you may even get to see yourself onscreen on the TODAY SHOW)... Walk a little westward and there is the Radio City Music Hall - home of the TONYs the Rockettes, and Hugh
There are many neighborhoods which are interesting to explore for historical or cultural reasons - there's the Museum mile on Fifth Avenue, the theatre district, the financial district ( close to Ground Zero) and not too far from a view of Lady Liberty, Greenwich Village, Southstreet seasport, or even a view of Manhattan island from the river via a ferry ride...No need to spend a lot of money if you are simply walking around ( you can probably go on one of the Hop On/ Hop Off buses to take you around and to get you back).
Plenty of inexpensive places to eat at - many choices of cuisine, many open a good part of the day and night..
Don't forget to see a show for the Broadway experience! And if you get a stiff neck from looking at the tall buildings too much - that is part of the experience,too
Enjoy!