What type of hunting? Upland birds, waterfowl, small game, big game, dangerous game, varmints, bow, pistol? Or are you still undecided, and basically just want to play with guns and maybe kill something now and then?
I think your options in NYC for anything related to guns are going to be about nil. The Catskills are probably your closest reasonable option where there are probably dozens of rod and gun clubs. I belong to a club further Upstate, and it's mostly a decent bunch of nice redneck types who love to chat guns and hunting and impress newbies with their vast knowledge and brilliant opinions. They have all sorts of recreational tournaments for shotgun, pistol, and rifle, plus just general use of the range to plink around. I pay $50 a year.
Can you picture yourself making long day trips into the Catskills to pursue this?
The New York State hunter safety course is required before you can get a hunting license, and it's actually a pretty good intro to the whole hobby.
Happy to help with more info if you're interested.
David