I live in the Jackson Hole area, and you've had good advice on the roads. Definitely avoid Hwy 22 over the pass from Wyoming into Idaho. We have just bought our first TT. Pick up next weekend from Pocatello, ID. We'll be driving home through Alpine, WY on US 26 rather than go over the pass.
There are lots of things to do in Jackson Hole, and it's a beautiful area. If rafting the Snake is not your speed, consider taking a dip in String Lake or at Colter Bay. Granite Hot Springs just south of Hoback is a popular campground with pull through spots. It's out a ways on a gravel road (so not necessarily a spot for a basecamp to see the area...though there are plenty of people who do that), but the road is good, and the hot springs are good for a soak. Plus lots of good hiking around there.
