Many suggest taking US93 across Nevada. From Spokane you could either take US395 southwest to I84, and I84 across Oregon and Idaho to Twin Falls, or I90 to Missoula and take US93 over Lost Trail Pass. US95 south from Spokane is also an option, though it involves dropping down to the Snake River level a couple of times. You could even stick with US95 all the way.
Looking at Google Maps, it appears that the shortest route is to take US95 (or US195) south to I84, then jog over to US93. It also suggests the all freeway route, I90 to I15 as the fastest (but not shortest).
Essentially you have a band of mountains from NE Oregon to Yellowstone that you have to either pass around, or cross. Once you've done that you have to traverse the Nevada desert, mostly along NS valleys, or skirt it in Utah (via I15).
Easiest and fastest route: From I-90 west (yes, west) take 395 south at Ritzville to I-82/I84, take 93 south through NV but take the 318 cutoff which joins back up with 93 north of LV. We took this route almost every year from north of Spokane to AZ for ten years.
2000 Born Free 24RB Class C
6.8L Ford V-10 Engine, E450 Chassis
2002 Honda CR-V toad
Roadmaster Sterling A/T towbar
VIP braking system
Eddyline Merlin kayak
I am boring, but I make this trip at least twice a year. We travel on I90 to I15 and on to Vegas. We sometimes take a side trip to Yellowstone. We dry camp at the rest area on the Contential Divide on I15. Then, we stay in Cedar City before going on to Vegas. In Vegas, right now, we stay at the Las Vegas RV Resort on Nellis just North of the Boulder Highway. We lived in the NW for 21 years and spend our summers there. We were raised in Vegas and spent the last 13 years before retirement there. We spend our winters in Vegas, SW AZ, SE CA, and Santa Maria, CA.