Hi! So you have some options.

The Salisbury area has access to the Taconics and the AT. Classic NWCT hikes like Lion's Head, Rand's View, and Bear Mtn are easy to get to from Salisbury. You also can dip into Mass/NY for great mountains like Brace & beautiful waterfalls like Race Brook Falls. Lodging here, whether Air BnB or the White Hart Inn, won't be cheap.

On the cheaper side and less rural would be staying in the Torrington/Winsted area. Here you can get to Haystack mtn and Campbell's falls in Norfolk, excellent views from People's State Forest, and you can head down to Litchfield for more trails at White Memorial (plus tons of great restaurants and shopping).

You could also consider staying in Kent, which is a beautiful town with some shopping and restaurants. It's a little less mountainous but still has great views by Caleb's Peak and Pine Knob, or head into Macedonia Brook state park.

Hope this helps!