My trips to NYC always revolve around eating, so instead of sightseeing I have good Mexican food, cupcakes, pizza, matzah ball soup, cannolli, bagels and lox... yes, Famous Fat Dave is my NYC guru.
But if you've never been, and don't want to spend the whole time gorging yourself, you must head to Central Park (esp this time of year, it should be lovely!) and if you're like me and enjoy trashy beachside proms, make the trek out to Brighton Beach and Coney Island.
Also, walking across the Brooklyn Bridge sounds like a cheesy thing to do, but it's really quite cool. Brooklyn's fun to explore too if you have time -- stroll down to Brooklyn Heights for the best view of the NY skyline, or go to Williamsburg and play hipster at the Galapagos Art Space or hit happy hour at my favourite bar The Abbey.
For penny pinchers, take the free Staten Island Ferry -- it takes you right past the Statue of Liberty. (You can pay one of the tour boats to take you out and go up it if you want, but I was never that pushed to do that. The ferry gets you close. And did I mention FREE?)
I have no idea about the Broadway shows or Empire State Building (heights, eeek!), I stay away from those touristy areas, but NYC is amazing off the beaten path.