You could get to Manhattan in a little over one hour from JFK by LIRR from Jamaica Station (take Airtrain from your terminal). LIRR will take you to Penn Station-midtown-34th St. E subway from Jamaica runs along 50th St. and then south on 8th Ave. The 42nd St. stop is close to Times Square. Most subway lines are accessible from Penn Station or just a few blocks away to take you anywhere. Just remember travel time back to JFK is at least one hour and you need to be back two hours prior to departure for check-in and security. Get yourself a subway map, the E and LIRR will get you to/from to Jamaica and Airtrain, the A (Far Rockaway only) will get you to/from Howard Beach where there is also Airtrain access to JFK.
You could get to Manhattan