airtrain brochure here
http://www.panynj.gov...-brochure-english.pdf
the system is free within the airport, and you pay to leave the system outside the airport. when you pay to leave, you can buy an onward subway or train ticket in one transaction, and the ticket machines take us cash and credit cards.
however, neither the LIRR or the subway will take you to Grand Central without a transfer en-route - if you've a lot of luggage, probably not a good idea.
I`d get the hotel shuttle bus. Airtrain will take you to Penn Station and you`ll have to get the subway. NY subway is horrible (like an unfinished tunnel with steel girders everywhere) imo. If there`s a few of you, you might as well get a cab $45