If you are staying near the airport, than the easiest thing is to first take the Air Train ($8.25) to Jamaica. If you are staying at an airport hotel, they have airport shuttles. Have them drop you off at Federal Circle so you can get the Airtrain towards Jamaica.
When you get off at Jamaica you have to pay $8.25 for the airtrain. Buy an MTA card from one of the machines and load $23 onto the card ($22 in fares and $1 in fee. This will cover your return on the Airtrain back to Federal Circle.)
While the subway is cheaper, I'm going to suggest you take the LIRR to Grand Central Terminal. It's much faster. From Grand Central it's a few minutes walk to Rockefeller Center, Times Square, and most things midtown.
But I'm going to suggest something no one talks about. It's what I call the "Poor Man's Sightseeing Bus." From Grand Central, walk over to 44th St. and Fifth Ave. From there catch the M55 bus south. It heads down Fifth Ave, makes it's way towards the the Staten Island Ferry terminal and returns up 6th Ave to 44th St. It passes the Empire State Building, the Flatiron building, goes through Greenwich Village, past City Hall, to the Ferry Building. It then turns north on Sixth Ave (Avenue of the Americas to tourists) past the World Trade Center site, back through Greenwich Village and Soho, passes a block from Macy's and will drop you off just south of Rockefeller Center. All for the whoppin price of $2.75. (It might be $5.50 if they make you get off and pay again to go north from the Ferry terminal.) It's paid with the Metro card. You can't hop off and on. It's regular city bus.
Of course, if there is anything else you want to see, you can always get off the bus and then get another one later. They run every few minutes.
I'm a native New Yorker and I recommend it to people. I've tried it myself as well. I got off at the Ferry terminal, took the free ferry to Staten Island and then caught the next ferry back. It's great on a nice day and you will get a glimpse of the Statue of Liberty as well as the skyline of Manhattan.
Be aware, this is the day after a three day holiday. If you try to get into Manhattan before 9:30 AM or back between 4 and 7 PM, you will be fighting rush hour crowds. That's also why I suggest the LIRR over the subway. It will be crowded but not as bad as the subway.
If you flying into JFK, and staying at an airport hotel, their shuttles can not pick you up from the terminal. Take the Airtrain to Federal Circle and then call them. The Airtrain is free between terminals and to Federal Circle. At Federal Circle, go down to ground level, turn right as you exit, and follow the awnings to the pick up area.
Airport shuttles are not that efficient regarding pick up and don't take you to the airport on demand. Usually every half hour so keep timing in mind. Give yourself at least an hour to get from Manhattan back to Federal Circle.
I've stayed at many of the airport hotels. Which one did you choose?