OP, frankly everything in Providence will feel very close after living in Boston. The question might be: which part of Boston would you live in if you could? When we moved, we knew we wanted a Jamaica Plain kind of vibe (walkable, shops, mix of people, parks, restaurants, still central to everything) and found it on the East Side of Providence near Pawtucket border. A lot of nice family neighborhoods people are recommending here would be far from retail shops and not very walkable.

FYI, I highly recommend Meeting Street Early Childhood Center (just south of RI Hospital). Great program, excellent teachers, decent price (compared to similar caliber programs).