I second everything Jayra has said about things to do and see in NYC as we live a couple of hours from NYC and have been many times for day trips and done pretty much the same things. We have a really good book of walks to do in different neighbourhoods but would definitely recommend all the usual touristy things first.
If you want to go further afield you will almost certainly need to hire a car whereas in NYC you almost certainly don't want a car.
If it would be a problem for you driving in the Manhattan traffic you can take the train, for example ,from Manhattan to Newark Airport to hire a car and then be on your way to places such as Philadelphia (2 hours), Washington DC (4 hours) or Upstate NY (2 hours).