That's a really good idea. If you are able to spend some time in Manhattan, then public transportation becomes a practical option. The subway from JFK to anywhere on the west side (A train) is easy as pie. You can't go wrong. Then, when you are ready, grab an E train on 53rd Street to Roosevelt Avenue in Queens, where you pick up a free transfer to the LGA bus (usually Q33, but Q47 to the Marine Terminal). The only problem with this plan is you have to stash your luggage somewhere while you wander around. I believe that Penn Station and the Port Authority bus terminal, both stops on the A train, should have baggage storage, but I can't speak from personal experience.

Bruce