Go to the junction of Routes 23A and 214 just west of Tannersville. You can approach this spot from Phonecia which is along Rt 28, about 12 miles south.
Now go about 300 miles northeast to Pinkham Notch NH and hike up into Huntington Ravine.
Seriously, there is nothing remotely resembling the "full exposure" of true mountaineering in the Catskills. A bunch of trails to snowshoe, and some short, delightful water ice routes to climb...but that's about it.