Don't go to Jamaica. Pretty rough neighborhood.

If you want to stay in Queens, look at Forest Hills, Jackson Heights east of Junction Boulevard, Woodside, Astoria, possibly parts of Flushing. All decent areas. Further west Floral Park is ok too.

Generally you don't need a car in the five boroughs of the city. You will need one in Long Island. Long Beach is fine but so are many other places in Long Island.

As far as being expensive – yes, the rents are high. Other than that, you can get great deals on almost anything you buy. If you are smart you can actually end up spending LESS money than anywhere else. Of course the converse is also true if you are not careful.