LaGuardia (LGA), Kennedy (JFK), and Newark-Liberty (EWR) airports serve the New York metropolitan area. The Port Authority of New York and New Jersey website has detailed pages for each airport which include sections on ground transportation options between the airports and Manhattan.
LaGuardia MTA Bus:
The only mass transit option between Manhattan and LaGuardia is the M60 route (PDF) of the NYC Transit bus system. It travels back and forth between the airport terminals and the Morningside Heights neighborhood, including stops close to GISS along Broadway at 116th St. and at 110th St. The M60 makes stops in Astoria, Queens, and in Harlem along the way. Although the bus schedule indicates the M60 should take 40 to 60 minutes to travel between the airport and GISS (depending on time of day), adverse traffic conditions can sometimes cause it to take up to 90 minutes.
The airport bus stops are green-framed glass shelters located alongside the ground transport road just outside terminal baggage claim areas. Cost is $2.75 and must be paid using a MetroCard or in exact change (bring quarters; dollar bills are not accepted).
JFK AirTrain and Subway:
At JFK, the AirTrain monorail circulates around the various terminals and then leads off to the Howard Beach stop of the subway A train, to the Sutphin Blvd. stop of the subway E train, and to the Long Island Rail Road's Jamaica station. The AirTrain costs $5, and then you have to pay the subway ($2.75) or train fare ($7-$16) to get to Manhattan.
Depending on whether you connect with the A or the E, the AirTrain plus subway combo takes 50-80 minutes to get from the airport to Midtown Manhattan. The A is more convenient in traveling to GISS from the airport as you would only need to make one transfer, but is 20-30 minutes slower reaching Midtown than is the E. An LIRR train from Jamaica station will bring you to Penn Station in Manhattan.
Newark also has an AirTrain service that circulates around the terminals and connects with the Newark Liberty International Airport Train Station. From the airport station, you can take a New Jersey Transit or Amtrak train to New York Penn Station in Midtown Manhattan. (Note that you pass through Newark Penn Station on the way, so be careful not to leave the train at the wrong Penn station.) The AirTrain itself is free. NJT fare from the airport station into Manhattan is $15, which is much less than Amtrak charges.
Airport Shuttles, Van Service, and Taxis:
There are various airport van, shuttle and bus services that connect the airports to Manhattan. Some have set routes, typically with drop-off/pick-up locations in Midtown Manhattan locations such as Grand Central Terminal (Park Ave. at 42nd St.), Penn Station (Seventh Ave. at 33rd St.), and the Port Authority Bus Terminal (Eighth Ave. at 42nd St.). See the LGA, JFK, and EWR ground transportation webpages for more info and for links.
City taxi cabs operate between Manhattan and JFK or LaGuardia. However, NYC taxis may only deliver you to Newark, and only Newark cabs can bring you into Manhattan from the airport. When cabbing in from one of the airports, be sure to get a cab from the taxi stand outside the baggage claim area so that you will be charged the city-specified rate. The livery cab drivers who may be trying to pick up fares in the baggage claim areas are likely to try to charge you more than a regular cabbie.
Cab fare between Manhattan and LaGuardia (LGA) is around $30-$35, and there is a flat fare of $52 between Manhattan and JFK airport. The fare to/from Newark (EWR) is around $60, plus a $5 surcharge if you are coming from the airport or $17.50 if you are heading to the airport. Add to the fare about $5 for bridge/tunnel tolls as well as a tip. Thus, you can see that cab fare can be quite expensive, particularly to EWR. Using one of the alternative mass transit or airport bus services may be slower and and less convenient but is significantly cheaper.
The NYC Taxi & Limousine Commission has more on their FAQ. (Note: Page did not include September 2012 fare increase info when last checked.)
New York City is located on Amtrak's Northeast Corridor. Trains stop at New York Penn Station in Midtown at Seventh Ave. and 33rd St. Travellers to and from Boston, Philadelphia or Washington, D.C., are likely to find that Amtrak is more convenient and probably less expensive than air travel.
Commuter rail service is provided by the Metro-North Railroad, the Long Island Rail Road, and New Jersey Transit. Metro-North connects the city to communities further up the Hudson River and in Connecticut, and terminates at Grand Central Terminal. Serving towns out on Long Island, the LIRR stops at Penn Station. NJT connects to various town in New Jersey and also operates out of Penn Station.