Teton County Library has quite a lot to boast about, especially for a small town library. There are quite a lot of free 'festival' days throughout the summer, many that will keep both kiddos and grownups occupied. Musicians are often brought in, and there are a lot of story-times throughout the week. The DVD section is fairly decent for a small library as well, and there's a garden that the kids can walk around to get some fresh air. Highly recommended. In the lobby you'll find a coffee/tea area for adults who needs a bit of caffeine to keep up with the kiddos - I believe it's a dollar, and it's on the honor system.
I actually live in Idaho but visit Teton County Library on a regular basis because it's such a great library. The kids section is wonderful. They have a large selection of pictures books and early readers as well as fiction for older kids and nonfiction on numerous topics. Is yor child really into a certain subject such as bugs or farm animals? There a bags available filled with 10 picture books on the subject so your child can read about his favorite bug to his hearts content. They also have computer access for older kids and a room with educational toys for younger kids.
The library also has great selection of adult literature as well. And the library also holds special nights for art viewing and literature groups for all ages.
The Jackson library is always on our list of places to visit while in the Jackson Hole area. They have story time Tuesdays, Thursdays and Saturdays plus special events. The kids area of the library is quite big and they certainly cater to the little ones. We were also amazed at the selection of Spanish-language books. Great for rainy days or for some quiet time.