The answer depends on whether it's your first trip to NYC or a repeat visit and your interest level as well as the weather.
On a nice weather day I love the double decker tour buses that allow you to take in large areas of the city. A walking tour of Central Park. A visit to the Central Park Zoo, a walk up 5th Avenue seeing the sites. I could go on and on. Basically have a good pair of walking shoes and head in one direction. If you don't like what you see turn and walk in a different direction.
If the weather isn't too good, my favorite is to take in a museum. My favorite is the Met, but the Guggenheim, Cooper Hewitt and many others are great choices. The NY Public library is awesome too.
Shopping--I have made 3 trips to Macy's in the last month and haven't scratched the surface! I have a Macy's in my local mall, but it cannot compare! It's 9 floors! If you love to shop, there's Macy's, Bloomies, Tiffany's, Cartier's and Canal Street! Don't forget the jewelry distric on 46th between 5th and 6th.
Broadway Show, go to www.broadwaybox.com and see what's being discounted and go to the box office with the codes to purchase the tickets. We recently saw Dirty Rotten Scoundrels with Jonathan Price and it was very funny and enjoyable. If you can get tickets to Wicked, it's worth every penny! I Love You, You're Perfect, Now Change is a cute look at married life.
Enjoy a good meal. Do some walking and read menus posted outside restaurants. Find one that interests you and take a chance.
Have you done the thrill ride at the Statue of Liberty? It's a blast.
There are so many things to do I could fill a book (and there have many written).
These are my favorites.