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If you need to stick with public transportation (There are buses between the two airports, they cost like $32 and it's a one bus ride full details of the options on this page -
http://www.panynj.gov/airports/ewr-a...nnections.html) you could do the following
Airtrain to the Newark airport train station
New Jersey transit to Penn Station in New York
E train to the Jackson Heights station
Q33 bus to the airport
Full details if the trip are in the Wiki with all the options explained in more detail than you will ever need
http://www.flyerguide.com/wiki/index...sportation_FAQ