you can either get touristy or get local. how long will you be staying? that makes all the difference
definitely check out the museum of natural history, the zoos, museum of modern art, etc. a lot of the rates to museums are suggested donations. hit up some broadway shows as well. locals don't usually go to these shows but whatever floats your boat...
learn how to read the subway map. download a subway app + yelp app. they will come in handy!
let me know what you're interested in... ie: nyc's history, electronics, art, food, etc. i'll try to come up with some tips.