The Bronx Botanical Garden is a place my father and I have been going to since I was a child. Here's a photograph I took of him here. It has hundreds of species of trees and flowers. Sitting in the middle of the Bronx, it's most recognizable as a green oasis from the grays and concrete of city life.
Its quite a trip out there but so very worth it if you wanna have the very special beach-experience! You can go there by car and then take the ferry either to Pines or Cherry Grove (I would prefer Pines!) or you do all by public transportation - there a "getaway tickes" for 36Dollars and the transfer works really smooth - may even be faster than by car!