If you're trying to avoid winter weather, it'll be easier to base your route close-in based on expected 1-3 day forecasts versus finding a route far in advance that's less likely to have snow. While there are definitely routes that are more likely than others to get snow, you'll be best served by being flexible on a routing and deciding whether it's better to head due west or due south based on forecasted winter weather.
In general, you're better heading due south out of Minnesota instead of due west in terms of missing winter weather/road closures in the Midwest. However, my gut feeling is that it'd be easier to cross the mountains on a more northerly track via Wyoming instead of a southerly track via Colorado. If going through Nebraska and having some time, I'd suggest going via US-30 vs. I-80. It's still Nebraska, but it at least goes through most of the towns directly and follows the railroad tracks, leading to a bit more interesting drive than I-80. It's also close enough to I-80 that if the weather isn't great or you need to stop for the evening it's easy to get to the amenities that an interstate route offers.
If weather allows, US-212 through South Dakota isn't bad either, though it's still not extremely interesting. It's more remote than I-90, though, and so I wouldn't do it unless the weather is good. The Great River Road is very scenic, but adds quite a few miles and pretty windy for a motorcoach. I also don't think you'd get any more advantage on winter weather than going generally due south over the prairie.
If I was making the drive, and the forecast allowed, I'd probably do I-35 (or US-71 or whatever's fastest) down to US-30 in Iowa. I'd then take US-30 west through Iowa and Nebraska, then take I-80 through the mountains to Utah. From there I'm not sure whether I-80 is better or dipping down to US-6/US-50 is better; I haven't taken either way via road and so I'm not sure which is better. (I haven't taken the Wyoming route either, but I remember going through Colorado being potentially treacherous in bad winter weather, so a northerly route seems easier.)