Amazin in Malden probably has the best tables in the area now that Brighton Billiards shut down a few years ago. Plus it's close to Boston. But as Ken said, it's a private club and with only 8 tables, I've seen it get filled up early with the regulars and you probably won't get much chance to play. I'm sure they'd let you play without the membership though for the day.
Snookers in RI is great and the place, can be a long drive from Boston if you are heading there during rush hour as you are hitting pretty much every major tie up to get there on 3 different roads hehe.
The Billiards Cafe in Ayer, MA where I play with my son most often is about 45 min from Boston, on Sundays there is a Junior league there that I'm always at with my son, doors open at 1, their league starts at 2, although the pool room technically does not open till 3 to the public. If you come by to play and see how the Junior program is, I'm sure we'll let you in. Send me a PM with a contact # if you like some info or to arrange something.
Stix and Stones is pretty good, I think it was called something else a bit ago, tables were well done, although not up to Amazin Billiards quality.
Come by Ayer Sunday, watch the kids shoot, there is also a snooker table there and a nice new 7" Diamond table in the back room to play on.