Yellowstone National Park
You can very easily drive from Jackson Hole to the Old Faithful Area and back in about 5.5 hours. I've actually done the entire park loop in one day, but it took about 14 hours (including getting out and walking around at scenic points of interest). Another fairly easy drive that would end up taking about 8 hours is to drive from Jackson to Old Faithful, walk around the Geyser Basin, get some ice cream at the general store and then proceed to West Yellowstone. There are some great restaurants in West Yellowstone for lunch (although since Jackson Hole doesn't have an Arby's, I usually just stop there!), and from there you will head south. You'll pass through some beautiful farm lands in Idaho and then head back over Teton Pass and back into Jackson Hole.
Grand Teton National Park
Head into Grand Teton National Park and the options for day trips are almost endless. You can enjoy a great lunch (and views of the Teton Mountain Range) from Signal Mountain Lodge, or hike around String Lake. Hidden Falls is a very popular hike that you can access by trail or by boat ride across Jenny Lake. From there, you can continue to Inspiration Point, or if you pack a picnic lunch, enjoy it in the shade of tall trees and enjoy listening to the water pounding on the rocks at the base of the Falls. Bison, elk and moose are pretty common sights in the park, but you should also be prepared to see black and grizzly bears if you are hiking.
Gros Ventre Wilderness
The Gros Ventre mountain range borders Jackson Hole to the East, and there are some great hike/bike trails, scenic areas and camping. Slide Lake is located just east of Kelly, and has a short and easy interpretive hiking trail that educates on the formation of Slide Lake, the destruction of Kelly, WY in the early 1900s and the geology of the area. Shadow Mountain is just north of this spot and is a very popular mountain biking and camping area.
Granite Hot Springs
Located south of Jackson Hole, this is a natural hot springs that has been developed with some great camping and food services. While the pool is built out, water is constantly flowing through it, and the temperature always seems to be perfect. The drive to Granite is beautiful, following the Snake and Hoback rivers almost the whole way.
Grand Targhee Resort
Located over Teton Pass in Alta, WY, Grand Targhee Resort has great family friendly activities and phenomenal views. Guests can play on the free frisbee golf course, or pay to take a scenic chairlift that offers gorgeous views of the West side of the Teton Mountain Range. From there, you can hike down, bike down or take the chairlift back down. There is also a ropes course, climbing wall and a restaurant/bar with great food and drinks.