If you're after a cheaper tourist-free (Yellowstone rapidly fills up after about 10am,the Upper Geyser Basin containing Old Faithful is truly hellish by about 2pm) alternative,we're recommending the drive south from Hamilton MT to at least as far as Arco ID,along US-93.As you approach the Idaho border,the road rises along the Lewis and Clark trail,then descends into the Salmon river valley,through the Salmon-Challis National Forest.
The scenery is pretty spectacular,every turn in the road (and there are plenty) opens another page of picturebook perfect views.
Before you reach Arco,looking left (east) as you're heading south,you'll hopefully see a 21-mile long fault line,caused by an earthquake in 1984 (I think).
The valley flattens out a lot towards Arco,but its worth going on to that town to see the conning-tower of the SSN-666,a decomissioned attack sub,erected in the roadside park.(Also a memorial dedicated to US Submariners).
Also,just behind it,is the Arco 'Number Hill',succesive classes graduating from the local High School go up there and paint their year number onto the hillside.
Take the opportunity to eat an 'Atomic Burger' at Pickle's Place,just across the road from the park.My butt didn't stop glowing or emitting noxious gasses the rest of the day :P
If you've gone that far,its worth going on to Blackfoot to visit the Idaho Potato Museum,which is more fun than it sounds (and you get a free packet of dehydrated potato if you're from out of state).
Between Arco and Blackfoot,you'll find the World's First Nuclear Power Plant,the EBR-1,which is quite interesting (admission free,open between Memorial Day and Labor Day).In the car-park there,you'll find the engines from,and story of,the nuclear powered jet-bomber which the U.S proposed to build in the early 60's.
I like Idaho.Lots to see and do.