As others have said, if speed and lowest elevations are your goals, I-15 is by far your best choice. Monida Pass, on the MT-ID border is your biggest challenge and the entire area around Butte is just darn cold most of the time, even in the late spring. (If I had $100 for every time I've frozen my ass off going through Butte, I'd be a moderately rich man.) I'd try and time things so you're going through Butte and tackling Monida in the afternoon if possible.
Do try and take in Yesterday's Cafe-A in Dell Montana, a weird little restaurant in a converted one-room schoolhouse. It's just off the freeway and a super-unusual place to eat.
If you're coming back through this area, the weather is good, and you want to explore some ideal S10 gravel roads, MT 509 between Monida and Hwy 20 to the east is absolutely fantastic.... 60-odd miles of good gravel traversing Red Rocks Lake National Wildlife Refuge. But this is very remote high country and you don't want to be back there without good weather, a full tank of gas, and some contingency planning.
- Mark