Rather than pay expensive taxi or bus fares from the airport, simply get the free shuttle bus from outside the terminal to Howard Beach JKF Airport station on the 'A' line subway train. Then take the subway straight to Manhattan. Cost? $2.
The free shuttle bus to Howard Beach (A train) or Jamaica (F) was discontinued when the JFK Airtrain started service in 2004. The fare is $5 one-way to and from your terminal from either station, on top of your journey into the city.
Subway rides are now $2 / ride but they're still the best deal in the city. Also, remember that, unlike London, one metrocard/subway pass can be used by several people as it is $2/ride wherever you go, so don't bother buying two $20 cards when one will do, just swipe it as many times as you need to!
The cost of travelling to JFK by tube is not only a single subway ticket, but an additional charge of $5.00 of the Air Train, which replaced the archaic bus system in Dec 2004, and now connects all terminals.