Just a note, your going to spend around 180.00 in tolls for the GW and Whitestone/Throgs Neck Bridges pulling a trailer. Not sure if you meant I84 or I80 from the Poconos to the GWB. For I80 that would certainly be the most direct route to the GWB (80.00 toll). But if you truly meant I84 I would cross the Hudson over the Mario Cuomo/Tappen Zee bridge (about 20.00) or better, the Newburgh-Beacon bridge (about 12.00) get off I84 at I684 south to I287 east then I95 south to the Throgs Neck Bridge. The latter route would be much less traffic as you would be bypassing the densest part of NYC.
If you are lucky enough to find an available campsite during any part of the summer grab it quick as Long Island camping is crazy busy.
Good Luck.