Sharon F&G has informal skeet and trap every Sunday at 10AM. You do NOT need to be a club member.
If you join that club you can use a hand-thrower any time (10AM to Dusk on Sunday, not sure starting time on M-Sat) on those fields and shoot skeet/trap to your heart's content.
I was a member from 1976 to 2000, so I can't speak for what's going on there since then, but they used to be very helpful to any new shooters and the Sunday shoots were for fun, no real competition so you could learn at a leisurely pace.
Massapoag Sportsmen's Club has registered Skeet and Trap matches all the time. MUCH more formal and didn't appear to be a place to learn (wrt shooting there on Sundays, only for the fast-paced experts). Very old info, but back when, they did NOT let you pick up your shells . . . once they hit the ground they belong to the skeet chairman who would sell them back to you! Not sure about their rules for during the week.
There is a Canton Club but I know nothing about it. Likewise there is a Stoughton F&G club that I also know nothing about.