I've never camped up there but I have camped in Beartown S.F. which is closer to Great Barrington, MA and not Pittsfield. The state campground in BearTown is open til end of October I think and is on a pretty little pond. It is also close to many hiking trails and the AT. October Mtn SP is also a bit north of Beartown and closer to Pittsfield. I've never camped there but I believe there are state campsites there too.
For a nice short hike with awesome views, there is a place called Laura's Lookout just south of Stockbridge, MA that has a small lookout tower on the top and a map pointing out various locations. It is short but there is a climb involved, but the reward is nice.
Jay
Jay