Welcome to the forum!
First of all, check out some of our winter driving resources.
For route, if the weather cooperates, I'd take the direct route over to Amarillo and then I-40 to Gallup. Then north through Shiprock into Monticello UT, or US191 parallels the same route on the Arizona side. Both are scenic and beautiful. From there, work your way up to Salt Lake and then I-84 and I-82 toward Seattle.
In Utah, check out Arches National Park and Canyonlands. There are many other great stops along this route as well, depending on how the weather is and your time frame. Happy planning!