Disclaimer: I know this is going to be a lot of info, but hopefully something will be useful for you while planning your trip.
I’ve explored quite a few of these roads since I moved to Eastern Idaho 2 years ago and the beauty never ceases to amaze me. I feel like most people take I-84 and completely miss out on all the amazing things Idaho has to offer, and that’s a shame. So, if it were me, I’d avoid that whole I-84 stretch. While the Blue Mountains are beautiful, there’s a whole lotta nothing out there from Baker City to Snowville.
Option 1
Coming from the Spokane areaI recommend jumping on Hwy 95 through Moscow/Lewiston and then Hwy 55 (at the Y) south to Boise via the Payette River Scenic Byway.
It’s a beautiful drive along the Salmon River and Payette River with tons of mountain views, lakes, and wildlife through the canyons, mixed with wide high mountain prairies and winding mountain pass roads. Even if you take the rest of I-84 from here, you’ll have packed a ton of scenery in to get you through the cow farts of Jerome. 
One of the other options from Boise is to take the backroads through the Sawtooth Mountains (which is some of my favorite territory in Idaho) by either:
A) Following Hwy 21 up through Idaho City to Hwy 75 and Stanley and Sun Valley. There is hiking, hot springs, and camping through this whole area.
B) Continue to Mountain Home, then turn north on Hwy 20 and follow that east through huge alpine valleys full of farms.
From there, you can continue on Hwy 26 to Craters of the Moon, and then get back on I-15 to continue on to Utah.
Option 2
The other option from the Spokane area is to make your way through Pullman (stop for some Wazzu ice cream and a tin of Cougar Gold cheese), then take Hwy 12 up through Lolo Pass. This follows close to the old Lewis & Clark Trail along the banks of the Clearwater River. There are lots of places to camp along this route, do some fishing, etc.
You can take a small detour up the Gold Rush Historic Byway.
After exiting in Lolo, you can head south on Hwy 93 through Salmon and Challis along the Salmon River Scenic Byway. Salmon is a sleepy old town, and the road takes you along the Frank Church River of No Return Wilderness. There is hiking, hot springs, ghost towns, and mining town all around this area.
https://visitidaho.org/travel-tips/enjoy-a-scenic-fall-drive-on-the-salmon-river-scenic-byway/