Chimney Mountain is a favorite.
It's a short hike to the top with two ridges two choose from. Although many may shoot straight for the actual Chimney on top, it is better to cut left near the top and end up at the false summit across the gorge from the actual spire.
Once you're there, you'll have the opp to get to the real summit, not by backtracking, but by dropping in some class IV choss that is very sandy so make real use of your hands.
The gorge is like some kind of huge-fern hide-out with a lot of deep cracks and pits to maneuver or avoid. I have spent a lot of time up here alone in winter but would not advise this on the first trip...
At the bottom of the gorge, you'll be compelled to start feeling your way around like a ninja as cold, almost paranormal air starts to fall against you from multiple directions. Close your eyes and put your hands out like for a minute and though you'll look a little yoda-like, you'll find the caves your looking for.
If your leading a climb, you can go straight up to top out at the right of the chimney but this is some genuine choss so don't start if you're not a NY trad leader who never falls or unless you like using slings on horns and behind flakes just to be old-school.
The base of the chimney is fantastic with a lot of little hide-outs including the roof that appears to be (and may very well be) held up by a single key-stone.
The chimney itself is an easy solo and you can drag a rope and rig the top if you're bringing guests. No shoes needed.
Between the chimney and the USGS summit is a phenomenal camping spot with no views but a lot of space in every direction.
This is the heart of NY moose country and the bear are everywhere that you, or someone else, might have left a bunch of awesome treats so don't be an idiot... You might be an idiot so I won't advise eating anything you find in the woods, but there seem to be an unlimited supply of wild berries on this hill and they can tend to make a man wander if he knows what he's looking for.
If you're heading south after, the best thing you can do for yourself is go to the Melody Lodge and get some wings (the best) and a pepper steak (order from the bar menu). They have mastered the art of vedging with fresh-clean paint on the Adirondack chairs and a stone porch with great views and visiting hummingbirds.
I reckon you can contact me via my profile if you want details of have specific questions.
Jim