AirTrain to Jamaica Station. Then, Manhattan-bound "E" subway to Queens Plaza station. Transfer to a Manhattan-bound "R" subway (on the adjacent local track). Take the "R" to 28th Street/Broadway station. Walk west on 28th Street to your destination, which should be just a block or two away from the subway station.
Note: If getting a seat on the "R" train is important to you, then consider transferring to the "R" at Continental Avenue/Forest Hills station, instead of at Queens Plaza. This can add about 15 minutes to your travel time since the "R" runs local and the "E" runs express, but since the "R" begins its run at Continental Avenue, you are virtually guaranteed getting a seat if you make the transfer there.
You should be able to obtain a free subway map at the Ground Transportation/Information desk in your terminal at JFK, or at the subway booth at Jamaica, just before you pass through the turnstiles. And at 28th Street station in Manhattan, the subway booth attendant should be able to provide you with a free Manhattan bus map.
AirTrain to Jamaica Station.