Rhode Island Audobon Society
A membership to the RI Audobon Society is great for kids. They have a bunch of kids related activities, like frog watching, story telling, meeting a variety of local birds and animals.
They have a small museum in Bristol where you can walk through and see animals, both land and sea, that are native to the area.
There are some great walking trails throughout the state where you can roam at your leisure. This a great way to just let kids be kids and explore the natural beauty all around us. Family memberships are very reasonable. You can also often find coupons on Groupon to get family memberships at a discounted price.
As a member, there are many events you can join, like easter egg hunts, and summer camps for the kids.