I think we've gone up to Idaho every which way there is, from SoCal and Phoenix, except 93 up through Nevada. I'd say, just based on experience, that the most direct and probably easiest route would be to take 40 to 93 to Vegas, and then up I-15 to 84, and then 95 at Ontario all the way up to CdA. The only two places it might get iffy are the Lewiston grade and the Whitebird grade on 95, if you're driving a moving van/car with trailer. A loaded car is fine, we drive up every summer with tons of stuff plus a roof pod and a big dog. 95 is a two-lane highway, so there will be oncoming traffic, and it's slow, but it is mostly in a straight line north. And there are some really beautiful stretches. It'll feel like coming home once you turn north on 95!
The other option would be to go up through Vegas and then 93 to 84 to 95, through NV, bypassing I-15 and SLC (which can be a traffic nightmare). We haven't done it yet because it is a pretty lonely stretch, all desert, and few options for overnight stays, but others here have recommended it over the years.
Going into WA is a third option. Two ways: (1) The fast way: 40-93-15-84 all the way to 82/395 in OR, then 395 to 90 through Spokane to CdA. (2) The very slow way: 95 from Vegas to Reno, then 395 (gorgeous, breathtaking, along the east side of the Sierras, but sloooow) all the way up to Pendleton, OR. Continue on 395 through Kennewick to Spokane, and 90 to CdA. We've done the entire 395 trip, and it was just wonderful, but it has to be part of a vacation, not just a means to get to CdA. I actually like all the nothingness from Kennewick to Spokane, nice rolling hills.
Whichever route you pick I may be able to suggest good places to eat and sleep. I wouldn't want to go 15 to Butte, and then west, either. We've done it twice. It's so far out of the way, and the freeway speed expectation is misleading: Summertime is roadwork time on MT freeways!