JFK Airport Shuttles

Fastest way:
Yellow cab (from JFK to any point to Manhattan) costs flat rate of $45 + bridge toll($5.50)
Cheapest Way:
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take subway line E to 7-th Ave stop .Cost of the ride - $7.25/person.
See http://www.mta.info/nyct/maps/submap.htm
for reference
Another way:
There is shuttle(look for it at Ground transportation counter or Welcome Center) that goes to any address in Manhattan for $18.50-$25 per person.
JFK_E

The trains(Metro-North trains) going there are out of Grand Central(~$10-off peak), the buses(Greyhound;~$15) are out of Port Authority Bus Terminal.
Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take subway line E to Lexington Ave stop. At Lexington Ave stop transfer to subway line 6 going downtown for one stop to Grand Central. Cost-$7.25/passenger.
For Port Authority Bus Terminal take subway E at Jamaica to 42nd street stop.
You can also take express NYAS(New York Airport Service) bus that will take you to either Grand Central or Port Authority Bus Terminal. The NYAS bus leaves in front of each JFK terminal and costs $15/passenger.
JFK_E

The trains(Metro-North trains) going there are out of Grand Central, the buses are out of Port Authority Bus Terminal.

Take Airtrain from any terminal on track 1 to Jamaica Station( be careful - on that track there are two alternating trains going to different routes; check the sign above the doors as the train arrives; the red colored sign is Jamaica one). Then take subway line E to Lexington Ave stop. At Lexington Ave stop transfer to subway line 6 going downtown for one stop to Grand Central. Cost-$7.25/passenger.
For Port Authority Bus Terminal take subway E at Jamaica to 42nd street stop.

He can also take express NYAS(New York Airport Service) bus that will take him to either Grand Central or Port Authority Bus Terminal. The NYAS bus leaves in front of each JFK terminal and costs $15/passenger.
JFK_E

There is door-to-door service that you can book at Ground Transportation(Welcome Center) counter after you arrive. It costs $18.50/passenger and takes between 50min to 1:30min to get to the destination(depends on if you being dropped off first or last - it is shared service).
There is also express bus to Times Square - it is called NYAS(New York Airport Service). It costs $15/passenger and it takes 45min to 1hr to get to Times Square.
You can find the bus in front of each terminal and buy ticket on the spot.
JFK_E