NY ATC hates giving VFR clearance in the area. Too much jet traffic to deal with. If you can get a word in edgewise, they'll often let you out over NJ (watch out for the Teterboro jets!) but I never see anyone heading out over Manhattan. You need to be above 2000' to be legal because of the height of the buildings and then you're starting to get into the altitudes the jets are using on approach to JFK or LGA.
Over the Hudson River (East River too) is VFR (uncontrolled) airspace up to 1500 feet [edit: this was changed to 1300 feet in 2009, after the accident referenced below] bank to bank up to the George Washington Bridge. My flight instructor told me that the radar shadow of the buildings makes it difficult to control, but I don't really know if that's the reason. You don't need ATC permission to fly there.
It's a zoo, though, pretty dense helicopter and small plane traffic, and everybody's looking at the big buildings, not the other planes. In 2009 a small plane and helicopter collided, killing 9 people. I've flown up there once, and it was totally worth it, but I probably wouldn't do it again.
That's not been my experience. I can regularly get my choice of the Hudson at 2000 or lower with a clearance (1100-2000 requires a clearance in most places). At night, I prefer lower as the buildings are more impressive when you're below the top of them. On a nice summer day, the VFR corridor can be a zoo, so I'll sometimes take the clearance at 1500 just to stay in less congested airspace.
If LGA is landing 13 or Newark is using 11/29, then you have a conflict with the airspace over the river, but it's very rare for Newark to use 11/29, and LGA traffic counts aren't all that high to present a conflict. Plus, if you're on a clearance and being controlled by LGA tower, they'll turn you if they need to to accommodate a departure or an arrival.
I often hear pilots complain they can't get a clearance through NYC Bravo airspace, but I've only once been turned down in probably 50 attempts. That was a Thanksgiving Wednesday when they had a radar outage and they were turning everyone away who wasn't landing inside the Bravo. I've found NYC tri-Bravo controllers some of the easiest and most accommodating approach controllers to work with.
Most people can learn to fly. It's a bit pricy but maybe not as bad as you think. Easier to do than drive a car, except for when you make a mistake. Look for a flight school nearby. The intro flight is usually at a steep discount.
There is absolutely nothing like flying a small plane. My first night solo (flew down the shore, over Atlantic City) is one of my best memories.
A couple HN posters contacted me by email, and here's some additional info I sent them in response to their questions specific to the Hudson River VFR corridor.
An acquaintance of mine manufactures a reasonably-priced gyro mount for in-flight photography and videography and is based out of NY. He has some excellent sample videos on his site as well: http://www.aerialexposures.com/video3.htm
Over the Hudson River (East River too) is VFR (uncontrolled) airspace up to 1500 feet [edit: this was changed to 1300 feet in 2009, after the accident referenced below] bank to bank up to the George Washington Bridge. My flight instructor told me that the radar shadow of the buildings makes it difficult to control, but I don't really know if that's the reason. You don't need ATC permission to fly there.
It's a zoo, though, pretty dense helicopter and small plane traffic, and everybody's looking at the big buildings, not the other planes. In 2009 a small plane and helicopter collided, killing 9 people. I've flown up there once, and it was totally worth it, but I probably wouldn't do it again.