I live in NYC too. And I'm fresh off of two weeks in Glacier National Park, so the the lack of mountains is killing me!
First, I'm dying to get out hiking this week and fight off mountain withdrawal. I'm in need of a partner too. you interested? I was thinking to go to the Adirondacks, but the Catskills, Whites, or even Harriman State Park are all viable options.
You can get to Harriman via public transport (Metro North) from Grand Central. The rest, not so much. I don't have a car either, but I car pool with those that do or rent a car and car pool with others that don't. It works out okay.
The best outdoor store in the City (that I know of) is Tent and Trails. It's at Broadway and Park Place. They lean towards heavier traditional gear, but sometimes I'm surprised at the UL stuff I find there.
Regarding climbing, I'm only just thinking about getting into it myselfso I'm not able to suggest much. My new co-worker is a pretty avid climber though. I'll ask him. I know most NYCers who climb visit the 'Gunks very regularly for great outdoor climbing though.
So, hope that helps. Good luck in the city. Let me know if you have any other questions or want to get out this weekend.