If your going to go through there in the winter stay to the southern route but you will still have some passes to get through to get there, weather can be sketchy.
November through march are the cool months, but feature snow birds. If you're looking to enjoy the state while here, October and March-may are warm and out of snowbird season. With that said, if you're considering a northern route like along the i40 you may consider October, or March thru may to avoid the ice and snow. A more southern route comes down to how hot you can handle.
I'll be traveling through with my 73 years old soon to be 74-year-old Dad. We will be limited to about 300 miles a day. He always talks about riding out West from Tennessee to California, so Lords will, we will take the trip this year. I think we can make it in 21 days. 9 days out, 9 days back and 3 days here and there. It's his dream to ride Highway 1 the Pacific Coast Highway. He just bought a New Ultra and he wants to go. It may take us a little bit longer but that's our plan. That's why I am concerned about the weather.
Well pick a time of year then choose a route accordingly. The i10 is boring as eff but clear all year long (dust storms aside). Coming through the us60 or i40 is much more pleasant but both feature high terrain and can get snowy.