Being spoiled from living in Berlin, I was a bit disappointed that a city like New York didn't have too much to offer when it comes to visiting fleamarkets. Considering myself as a junkie in this area, I recommend visiting cities in New England for finding affordable antiques and other accessories. However, if you don't feel like traveling for a piece of furniture or other classics that might stay in your apartment for decades, try visiting your neighborhood Salvation Army. I checked out a few in Manhattan and Brooklyn and I found the best and biggest one to be on 48th Street between 10th and 11th Avenue. But there is still hope to find more stuff on actual flearmarkets like the "Antiques Garage" on 25th Street between 6th and 7th Ave every (see pic below) and in the East Village on Avenue A and 11th Street on Saturdays and Sundays. The first one has over 100 vendors and can be found on two levels of an old parking garage in a Chelsea.
Be sure to be kept posted on more info in the future..