Just a few points about the various routes available to you. The rules-of-thumb that Michael gave you (All routes are subject to adverse weather. Elevation matters. Preparedness of local road crews matters.) are all right on. There are others such as that the shortest route is generally best because it minimizes your time on the road. But as with all such generalized rules, there are exceptions and details to be considered.
The difference between the I-80 and I-70 routes is really minimal in terms of miles, 25-50 miles depending on how exactly you get to the two highways. One of the main winter driving rules-of-thumb is to stay on the Interstates which are the straightest, flattest, and get the most attention. That would be I-35 straight down to I-80 at Des Moines and then either continue on to Salt Lake City or take the I-76 cut-off to Denver and I-70. It's only another 25 miles to get use I-40 IF you use US-400/US-54 from Wichita to Tucumcari NM. That gets you farther south if there is snow forecast along I-80/I-70, but violates the 'stay on Interstates' rule. I will say that I have driven US-400/US-54 and its a decent road, two lanes but relatively high speed (around 60 mph) with only a few slower limits as you go through the widely scattered farming communities.
So, if the weather cooperates, it really doesn't matter much which route you take, so you might as well choose the one that has things along it that you'd like to see. Just one example of a sight to see which you can only see along one of the routes: I-80: the Great Salt Lake; I-70: Arches National Park; I-40: the Grand Canyon. Your toddler and cat won't care which way you go, so you might as well plan to make the adults happy.
Plan also on four full days for the drive no matter which way you go. That will leave you a good half day to sit out any weather you don't feel comfortable driving in. If you need to sit out longer - do so. For that reason, and because even though you plan on one route you shouldn't make your final decision until you get a goo long range forecast just a couple of days before departure, you shouldn't make reservations. There will be plenty of vacancies at this time of year, and you'll never be more than 25-30 miles from a good warm motel anywhere along any of the routes.
Mostly, just plan ahead but be flexible.
AZBuck