The absolute easiest way to get to JFK from Manhattan, or from Manhattan to JFK, is to take the Long Island Rail Road (LIRR) out of Penn Station to the Jamaica station. Almost every LIRR line goes through Jamaica, but a few don't, so make sure you're getting on the right train. It's 35 minutes on a LIRR train to/from Penn Station. There are many trains that run between Jamaica and Penn Station every hour.
From Jamaica, it's a 30-second walk to the JFK AirTrain, which takes 5-10 minutes to get you to your terminal. The AirTrains run every couple of minutes.
I had allowed myself 90 minutes for the trip to JFK from Penn Station, and it probably would have taken that long by taxi or by subway. It ended up being 45 minutes from the time I bought my ticket at Penn Station until I was walking into Terminal 4 for my outbound flight. Total cost on a Sunday: $10. (It's a total of $12 if you travel during peak hours.)
A cab would have set me back more than $50, and would have taken at least the same amount of time and probably more. It's AirTrain and LIRR for me from now on when landing at JFK.
Penn Station -- LIRR to Air Train