If you are wanting to get a feel for the country and have 8 days to 10 days to travel around, I would make a big loop. Fly into Boise, head north to Riggins -> Lewiston -> Coeur d'Alene, ID -> Missoula, MT -> Bozemen, MT -> (through Yellowstone) West Yellowstone, MT - > Idaho Falls, ID -> Twin Falls, ID (say hi to Ray_G ;-) ), maybe swing up to Sun Valley or head on back to Boise.
That loop would probably take 20 hours of driving but could easily be spread over your time period. At a leisurely pace you'll get a great feel of the larger cities and small/microscopic towns as well as the variety of topogrphy the state has to offer. As a bonus you will see some of western Montana and Yellowston National Park.
If you wanted to cut some time off the loop, there is a town called Grangeville between Riggins and Lewiston. From there you can head east along the Clearwater and Lochsa Rivers over to Missoula. A beautiful winding drive but you better take Dramamine if you get car sick.
There is plenty of beautiful scenary, fishing, and hunting along the way to keep most people entertained. During the early fall would be a great time to make the trip. The temperatures should be accomodating without the weather hassles of some of the other seasons. Unfortunately, many of the incredible colors of spring and summer will have turned brown by this time.