Here's more detail. When you arrive at JFK, follow the signs to the AirTrain, and board an AirTrain headed to Jamaica Station. (When the train pulls into the station, the sign will indicate its destination, usually accompanied by an announcement.) Jamaica Station will be the last stop on the train.

When you leave the train, you will have to buy a ticket, called a "metrocard," to exit the station. That's when you want to follow the directions that guv1976 provided. Buy your metrocard and use it to exit the station, but then save it as you'll need it for the subway and the bus.

After exiting the AirTrain station, follow the signs to the subway, and then use your metrocard to enter the subway station. Look for the signs for the E train headed in the direction of Manhattan, and take the E train to the Jackson Heights-Roosevelt Avenue /
Broadway stop. Exit the subway, go upstairs, and follow the signs to the Q70 bus, which picks up on the street outside the subway station. (When you come up stairs and go through the turnstyle to exit the subway, turn left to the street to catch the bus.)

The bus doesn't make any stops until it gets to the airport, but then makes several stops at the airport at the different terminals.