If you want real non-touristy things, I’d say go to a smaller museum such as the Museum of the City of New York. Walk a lot and take the train and just explore the different neighborhoods, and discover the little quaint shops and boutiques sprawled around the city. Take the GWB (George Washington Bridge) and check out the Palisades. You could hang out in Central Park, but Riverside will have way less tourists. Harlem is a good choice, plenty of good restaurants, especially if you want to venture out into more world/ethnic cuisine. Also, you could head over to Brooklyn and get some of Junior’s cheesecake (kinda touristy, but not as much as doing most things in Manhattan), go to the Prospect Park Zoo, see if there’s a concert or a performance in a park. Hang out in the trendy parts of Brooklyn. Most non NYC people forget that NYC is so much more than Manhattan. Go to the botanical gardens in Queens, or Jones Beach. There’s plenty to see aside from the touristy areas. I’m from NY, most of us haven’t even seen it all and like to avoid the tourist places.
And definitely get a slice of pizza, standing up, extra garlic (if you like that). A bagel and a dirty water hot dog is also choice too.