The PJCC grass courts were the site of my S&V upbringing (thanks to parents' membership). They also had the wisdom to add a pair of clay courts there a few years back, too. Beautiful place, but the tennis has always been a lesser priority there compared with the golf.
To the best of my knowledge, the Warwick Heights club is also members only. The clay courts at Roger Williams Park are probably the best bet when they're in business (as far as getting onto some clay), but there are lots of nice parks, etc. with hard courts scattered all over the place. I'm in southern RI and coach the teams at North Kingstown.
My e-mail: firstname.lastname@example.org
I string racquets (usually overnight) and also play a good bit in case anyone needs a hand in one area or another. Since I'm currently shopping a new computer, my response to any e-mails may take a few days, but I'll do my best to keep tabs.