Bradbury Mountain in Pownal is a wonderful spot that is treasured by the locals. There's a short hike to the top of a small mountain that has surprisingly good views. It's the perfect place to introduce little kids to hiking. You'll find wooded trails that lead up to an exposed peak with great rock-hopping opportunities. Even the toddler crowd can get to the peak with little adult help. We usually take the Summit Trail straight to the top, then one of the other longer trails back down. Perfect spot for lunch with a view. There's a nice playground, too
This state park is also well-known for it's mountain biking trails. Many of them are advanced trails, but they are also has some good trails to introduce young kids to mountain biking. Most of the mountain biking takes place across the road on the east side of the park.
Resources: