A.R.
I don't know which town you're in, but we moved to Bristol a couple of months ago, and the playground at the Common is very nice; also, if it's a really nice day, there is a small playground down near the Bristol Town Beach at Colt State Park, a good option if you want to do the beach & a picnic, too.
We drove past the Roger Williams Park playground one day after we went to the zoo, it looks like fun, but we are waiting to go there on a weekend so we can do the carousel, too.