Looking at the places you want to see, and assuming you're coming from Burlingame CA, I would take your favorite route to 395 and follow that up through the desert to Route 20 and go east for a bit. Near Burns, take 205 south and check out Steens Mountain and the Alvord desert. You'll pass by the Malheur Wildlife Refuge, scene of our recent infamous "uprising".
Get over to 95 north and travel up through the Jordan Valley area. I don't know if I've ever felt more in the middle of nowhere than there. It's a great ride through an incredible canyon.
Continue up 95 and you can keep going up into Idaho at that point, but I wouldn't miss Hells Canyon. You can get off 95 around Cambridge and head up 71 to the Brownlee-Oxbow Hwy. The ride is really cool all along the Snake River. There's a campground right on the river, Copperfield CG. Now you're nearing Hells Canyon itself, including the dam. Here's a link you can check out for that trip:
Driving Into Hell ('s Canyon) - Brownlee and Hells Canyon Dams, Scenic Byway
If you're feeling particularly brave, you can get back to the Idaho side and take any number of **very small** roads to get back to 95 north. You'll be really in the middle of nowhere here and will want to bring actual detailed maps; GPS may not help you out much, if at all. I lived out there a long time ago and there wasn't any GPS. But it's really the most beautiful high desert country and a time I remember fondly.
Once you're back on 95, you can choose your scenic routes from there. I'll let other folks more familiar with those areas chime in with upper Idaho and Montana routes. Good luck and have a great time! Four weeks is a really nice amount of time to put toward this journey.