To answer your question......
It is a pretty hike to the Snow Hole. Not a great deal of elevation change, but there are some nice vistas along the way. Much of the trail from the Petersburg Pass to the Snow Hole is part of the Hopkins Memorial Forest, which doesn't allow bicycles.
The Taconic Crest trail south to Berlin Mt. is a bit chewed up from ATV/motorcycle use. I'd highly recommend the stretch from Mattison Hollow, east up to the TC trail and north on the TC to Berlin Mt. as a much better way to go - albeit longer. I have seen mountain bikers on this part of the trail and the trail is in good shape.