Trains leave from Grand Central Station in NYC to New Haven about every half hour during the day. The cheapest way is to take Metro North. You can check out schedules and fares here:
The New Haven railroad station is called Union Station. From there, a taxi can take you to campus and/or to your hotel (a 5 minutes ride).
The closest airport is Tweed airport in New Haven. However, this is a small airport with few flights. From here, a 10 minute taxi ride will get you to campus and/or your hotel.
The next closest airport is Bradley, which is an international airport. From Bradley there are shuttles and taxis to New Haven.
Connecticut Limo has a regular shuttle service from Bradley to New Haven, but not door-to-door.
If you are coming in from any of the three New York airports (JFK, LGA or Newark), you can use the shuttles as well. The two operating shuttle services are Go Airport Shuttle and Connecticut Limo. There is also an Amtrak train from Newark airport directly to New Haven Union Station.