I've done this hundreds of times.
Don't go through Manhattan, go through Staten Island.
Take the Parkway north , exit right after the Raritan/Driscoll Bridge (I think it's 129).
Take the Outerbridge Crossing.
Get onto 440 towards the Staten Island EXPWY.
Get on Staten Island Expressway, stay straight until the Verrazano Bridge.
Go over the Verrazano, get onto the Belt Parkway take it straight until the airport exit.