I have taken the subway and AirTrains quite a few times on some trips last year to JFK. I thought it was pretty easy overall, but here are the tips I'd give to someone using it for the first time.

I am assuming you'll arrive on UA Terminal 7. You have to go outside and cross the street to get to the AirTrain station, which runs on an inner elevated loop around the JFK terminals. Take the elevator or escalators up to the AirTrain departure level. Look at the LED signs and listen for the AirTrain going to Jamaica station and hop on that one. This will take you to the subway.

They were currently doing a lot of work to the AirTrain/subway station when I was going through last year to make it easier to connect. This is the best way at the time I was doing it but things may have changed. Stop and pay your AirTrain by buying a MTA fare card at the vending machines. You might as well put more money on the card since you'll need the card to pay for the subway as well. MTA gives you a 20% bonus when you put $10+ on your card.

Exit and take the elevator/escalators out to the street level entrance. Take a left exiting the doors and you will walk along a sidewalk going under the LIRR tracks. The next corner is the entrance to the Sutphin Blvd/Archer Ave. station. Make sure you have your MTA card and go through the turnstiles and make sure you are headed toward World Trade Center/Manhattan and on that platform.

Returning to JFK, AirTrain heads toward Terminal 1 first and then stops at every terminal till it gets to T8/9-counterclockwise. The quick way is to get off at T1 and walk straight across 25 feet and hop on the AirTrain terminal loop train that goes clockwise. The next stop for that train is T8/9.

Note that the AirTrain says Jamaica station but it really goes to the Sutphin station, the stop before Jamaica. I think it is extremely confusing.

You can Google MTA and AirTrain sites for more info.

Happy travels.