I'm seeking recommendations on the best places to camp in Western MA. Preferably not busy campgrounds, and with access to good hiking.
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All public campgrounds in western MA are going to be "busy" on weekends in the summer. In fact, I wouldn't even try finding a campsite later than afternoon on Friday or on Saturday.
As for hiking, look at maps to see what the hiking opportunities are by each campground you are considering. A good place to start is with the state-wide trails map. https://www.mass-trails.org/map/ma.html#map=10/42.38746/-72.83798 is zoomed in on the western part of the state. Zoom in more, and you'll see individual trails.
By the way, that web site was created by a member of our Town of Groton Trails Committee. It was modeled after our very successful interactive trails map that he also created, expanded to the whole state. The data is harvested from OpenStreetMap, but displayed in a trail-centric way.
After some poking around on that map, you'll see there are many many hiking opportunities all over MA. Even in just the "hill towns" area (from the NY border east to the edge of the Connecticut Valley), there is a wide range. This includes the popular AT and areas around Mt Greylock, to much more obscure woods walks and other local hikes of all types of terrain.
Off-road vehicles are allowed in some places, like many of the trails in October Mountain State Park. As a hiker, you might want to avoid those areas. There are still plenty of trails what won't have dirt bikes and the like on them (at least not legally).
This is on the eastern side of “western”, but Tully Lake in Royalston MA has tent-only camping (hike/paddle in, no vehicle access) and trails beside a small gem of a lake. http://www.thetrustees.org/places-to-visit/places-to-stay/tully-lake-campground/. It’s also the site of a great trail run / mountain bike / paddle relay triathlon each October which draws plenty of lycra and carbon fiber but also blue jeans and aluminum camp canoes.