By Air:

A good search engine for cheap flights is kayak.com . If you specify HVN as the destination airport, the engine will show you flights to all airports within 100 or so miles, this includes all the New York area airports, Bradley in addition to New Haven. If you specify BDL, you will also see fares into Providence, RI.

Tweed New Haven Airport 
Nearest airport, 4 miles from Yale Campus, is Tweed New Haven Airport (code HVN). US Airways flies into New Haven from Philadelphia to where connecting flights are available. The most convenient way to get to campus is a short inexpensive cab ride. There are taxis available there, or you can call one at 203-777-7777. 
Flying into New Haven may sometimes cost a little more, but you should consider that you will save yourself $30 or more each way in ground transportation, this is without counting the value of convenience if you can practically walk off the plane onto our dance floor.

Hartford-Springfield Bradley Airport 
Bradley International Airport in Hartford-Springfield (code BDL) is about 45 miles up I-91. 
There is bus/minibus service to New Haven from Connecticut Limo. Upon request CT Limo will drop you off at Phelps Gate on the Yale Campus, close to Fest Locations. Reservations may be necessary for CT Limo, and if multiple people travel together (both ways), companion fare tickets are half-price. Call them for info. 
Other limo options are listed here . 
There is no train from the airport. 
You may consider renting a car, Friday-Monday from Enterprise at BDL airport starts at $99, so this is more advantageous for 2 or more persons. You will not need a car during the Fest as everything is within a few blocks. 

JFK Airport and La Guardia Airport, New York 
Airport Bus from JFK or LGA ($15) to Grand Central Station in Manhattan, MetroNorth train to New Haven Union Station($13). From train station to Yale, see here . The airport bus goes every 20 minutes, the train every 30 minutes, and in between you can check out Grand Central Station which has great food and bookstores. This itinerary is faster, more interesting and cheaper than a limo/bus. 
Prime Time Limo will drop you off at Phelps Gate on the central campus, 2 blocks from Yale Tango Fest locations. 
Connecticut Limo ($65 O/W) (van or bus). Upon request CT Limo will drop you off at Phelps Gate on the Yale Campus, 2 blocks from Fest Locations. Reservations may be necessary, and if multiple people travel together (both ways), companion fare tickets are half-price. Deals differ per airport, call CT Limo for info. 
The limo/van/bus option sounds convenient but may involve waiting at the airport and multiple stops and drop-offs between the airport and New Haven; this could easily add up to several hours.

Newark Airport, New Jersey 
The Airtrain will take you from your airport terminal to the Amtrak station just outside the airport. Many (not all) Amtrak trains on the Northeast route stop at Newark International Airport. When searching train schedules, make sure the departure station is Newark International Airport EWR (not Newark NWK). Book a direct trip, so you don't have to change trains in NYC. $34 to $54 O/W depending on the train. Holders of a Student Advantage Card get a 15% discount (must book at least 3 days ahead of travel). 
Connecticut Limo ($80 O/W). Upon request CT Limo will drop you off at Phelps Gate on the Yale Campus, 2 blocks from Fest Locations. Reservations may be necessary, and if multiple people travel together (both ways), companion fare tickets are half-price. Deals differ per airport, call CT Limo for info. 
The Amtrak option is far more comfortable and cheaper than CT Limo.

By Train:

Metro North 
MTA-MetroNorth commuter train from New York Grand Central Station to New Haven Union Station. The preferred choice from NYC as off-peak tickets are only about $13 O/W. Travel time is 1.40-1.50h. MetroNorth trains are only 10-20 minutes slower than Amtrak but cost less than half. However if you are already on an Amtrak train from somewhere else going through New York, it is better to stay put as you will not save any money by changing trains, and it is very inconvenient to travel 15 blocks between train stations in Midtown NY. 
From train station to Yale, see here .

Amtrak 
Amtrak serves New Haven Union Station (NHV) every hour or so, from Boston, Vermont, New York Penn Station, Washington and points in between. Trains usually have electrical outlets and a bar/snack car. Click the link to search train schedules. 
Holders of a Student Advantage Card get a 15% discount. You can sign up for a Student Advantage card online ($20 or $25) and get your membership number the same day. To get the discount at Amtrak you must book your ticket at least 3 days ahead of travel.

From the train station to the Yale Tango Fest locations 
From Union Station it's a +/- $5 taxi ride (203-777-7777) to Yale Tango Fest locations. If you see a Yale Shuttle bus (looks like a white and blue schoolbus) idling in front of the station, ask the driver if he stops near GPSCY's (or Harkness Lounge/Dorm), and tell him you're a visitor to Yale; show him any university or student ID you may have. The Yale bus is free but you may need to persuade the driver of your connection to a Yale event. The distance is walkable but this is not recommended as the route goes through a confusing public housing project neighborhood.

By Bus:

Greyhound and Peter Pan have stops at New Haven Union Station. 
Holders of a Student Advantage Card get a 15% discount. You can sign up for a Student Advantage card online ($20 or $25) and get your membership number the same day. 
From train/bus station to Yale, see here.

By Car:

New Haven is 1.30h from New York City, 2.15h from Boston; 1 to 2h hours from Western Massachusetts, 2h from southern Vermont, 2.30h from Albany NY, 3.30h from Philadelphia and 5h from Washington, DC. 
New Haven is at the Junction of I-95 (exit 47) and I-91 (exit 1). These exits (Downtown New Haven / Rt 34) will take you onto the Richard E. Lee Connector, after 1 mile take exit 2 or 3 (see below). The Yale Tango Fest locations are all off York Street.

Directions to GPSCY's, 204 York Street: Take exit 3 on the connector and take a right at the light (at the Walgreens) onto York Street. The location is off York between Chapel St (3rd light after the Walgreens) and Elm St (4th light). Walk left onto the little street by 202 York (Yale Daily News building). GPSCY's/Gryphon's Pub is in the back by the arch. Use the parking meters on weekdays and Saturday until 7pm. After 7pm and Sunday, no meter needed. The nearest Yale Parking lot that is open to the public on weekends is on Temple Street half a block north of the New Haven Green. 
Central campus map