The Travel Editor replies: The cheapest way into town is to take the free shuttle bus (ask at information) to Howard Beach and then take the subway from there into central Manhattan ($1.50). The journey should take just over an hour.
The most conventional way to get into central New York from the airport is by yellow cab. This will cost about $30 to $40, depending on the traffic (you should allow about an hour for the journey). Make sure you insist that the meter is switched on (private deals rarely work out cheaper).
Another convenient and cheaper option is the Super Shuttle (a shared cab). These can be picked up outside the airport, by the yellow cabs, or book one from the information desk (next to baggage claim). A taxi in the shape of a blue van will drop you and your fellow passengers at your destination for about $15 (midtown), and you can book it for your return journey.