The route you stated that you are planning on taking....out of Cheyenne, WY....north on I-25 to Billings....then west on I-94 is an option.....but likely a potentially.....bitter cold one that time of year and there is NO way of avoiding mountain passes along the way.
Your other alternative would be to continue west on I-80 out of Cheyenne and pick up I-84 in Utah......but there are also mountains to take into consideration just east of the Salt Lake/Ogden area that get a lot of snow as well........but if all goes well past that area......its wide open as far as passes go until you reach Snoqualmie......assuming you head north on I-82 out of northeastern Oregon which, BTW.....has some elevation rise through the Blue Mtns. that could get tricky......or, like another poster said......you could stay on I-84 through the Columbia Gorge into the Portland area.....then head north on I-5 to Tacoma......without any passes to worry about......maybe ice in the gorge though.
Any way you look at it though.....travelling through the West that time of year.....CAN mean trouble with various winter weather conditions, blizzards, etc. to think about......and carrying chains IS advisable as well as monitoring the weather forecasts on a frequent basis!
Good Luck, be prepared and enjoy the scenery along the way.
