In the winter, other than steelhead (Salmon River area north of Syracuse, and streams in western NY and PA) most of the fly fishing is heavily weather dependent or localized (power plant outflows in some saltwater spots etc).
Most of the trout fishing won’t really get underway in earnest until the water temps start hitting 50 degrees and the spring hatches start in early/mid April. But there are some year round trout fisheries that can be quite good as they are tail waters so water temps stay consistent:
NY East Branch of the Croton River, about 1 hour from NYC
And in CT there’s the West Branch of the Farmington but it’s more of a “schlep” (just testing your NYC vocabulary)
A good source of info for you is Urban Angler, a fly shop on 206 Fifth Avenue (between 25-26th Streets) 3rd floor Urban Angler Fly Shop | New York City | Arlington Virginia | Washington DC
There are also some fly fishing clubs in NYC. It’s a great way to break into the NYC fly fishing “scene”, meet some new fishing buddies and learn some new water.
Theodore Gordon FlyFishers with Tuesday lunches, conservation activities, group events TGF
NYC Chapter Trout Unlimited NYCTU Home
as well as other chapters in the greater NYC area (NY, NJ, CT)
A very helpful book is Eric Newman’s Fly fisher’s Guide to New York for a good description of waters within striking distance of NYC including the Catskills, downstate NY streams like the Croton, CT streams like the Farmington and Housatanic, and streams in NJ, LI and some mention of shore based saltwater opportunities.