From Grand Central (someone else hopefully can guide you to Grand Central), take Metro North to New Haven (end of the route so have your son pick you up here)
You can get a taxi or private car to Grand central Station.
You can also take a subway / train.
When you say arriving in New York City--that really means Kennedy .,Lagradia , White Plains..none of which are actually in the New York City --though they are listed as being so....
Depending on which airport you arrive at --will determine which transport to the actual city at grand Central Station would be best for you.Edited: 3:06 pm, June 29, 2013
You will probably spend more time commuting from the airport to Madison than on your trip to NYC and hopefully you will arrive early in the day... you will need to take the subway to Grand Central or Penn Station to pick up the train into Union Station in New Haven CT and transfer on Shoreline East to Madison. It may be easier to take the CTLimo.com airport shuttle, which also stops at Union Station and take the train to Madison from there or have your son pick you up in New Haven which is less than a 3rd of the distance to when hes picked you up in Hartford.... taking the train from the airport will make for a very long day!