>>1115751What exactly are you looking for in an /out/ experience?
If you're looking for backpacking and places to stay overnight, there really isn't anything in the city or the island. You'd have to head up north or to NJ for that. Long island has some decent trails you could go on a day trip to and you could take the LIR out there depending on what you want to do. I don't know about getting around without a car outside of the island since I haven't really been.
Your best option (and a decent one at that) is Harriman to the north. There's plenty of trails up there and has been my go-to when I want /out/. Search myharriman for some good info on taking the train up there. I actually just got back today from backpacking out there and even with the rain I had a great time.
Try looking up the NY/NJ trail conference website for nearby trails as well. The only trail that I really know of in the city (and plan on doing completely sometime in the future) is the long path.