You are going to need to use MetroNorth train, which is the commuter train that runs from New Haven directly to Grand Central.
Amtrak is the passenger train which runs the same route, but ends up at Penn Station.
Your best bet is to drive to the
New Haven Station on Union Street.
The link provides all the info you should need to get there, and phone number for parking details. I have been there, and parking is tight, but reasonable.
Here are the links to the MetroNorth website, along with two maps of the station stops for MetroNorth.
http://www.mta.nyc.ny.us/ http://world.nycsubway.org/us/metronorth/mnrrmap.html http://www.mta.nyc.ny.us/mnr/html/mnrmap.htm
You may notice that Waterbury is closer for you, however, I do need to recommend that you go to New haven. Waterbury is my home town, and, needless to say,I no longer live there. Drive thru on the highway, but be careful where you get off.
Dont really have any personal info on NYU, here is the link for the campus and surrounding area.
http://www.nyu.edu/about/campusinfo.html