I agree with everyone's winter warnings, and Northern California and Oregon are probably the best choice to tackle the snowy mountain part, but don't be surprised if it's difficult getting through the mountains. I have been stuck in snowy mountains on interstates 3 times during winter driving out west. In Idaho I managed to pull off to the nearest hotel, ditto once outside Aspen, Colorado ($$$), but another time in Colorado when the snow came so fast that all vehicles were snowbound overnight along the interstate. A family in an RV invited me and my friend inside to sleep the night (lucky!). They turned out to be metal detector aficionados and showed us their display cases of finds from all over. Which leads me to my second bit of advice: be open to side adventures, serendipity, eccentricities.
And check the timeline on this one. Will there be a return trip too? It's hard to imagine making it with any less than 3 weeks if you are taking the southern route.