Hi,
Where about's in New York, Manhattan? If so, I feel that the only reasonable place for GA to fly out of is Essex County Airport, known as Caldwell in New Jersey. It's a bit of a hike to get out there (1 hour bus and walk, 30 mins by car). If you want directions for the bus route (you will definitely need these) - feel free to PM me.
Teterboro is nearer but more expensive, and you spend too much time waiting for clearance.
At Caldwell, there are 2 FBOs, Caldwell Flight Academy and MacDan. I have only experience of CFA. They are fine, if a bit inpersonal. Rates are roughly $77 per hour for a Warrior/C172, $100 dual. You will be asked to pay for Fuel Checker thing, so bring your own.
If you've only got a couple of weekends, strongly suggest going for a jaunt down the Hudson VFR Corridor, as a must do flight.
Cheers
KCDW