The way my current contract is going, it looks like I’ll be getting some down time from the middle of November until possibly after Christmas. I’ve been toying with the idea of asking my employer for a leave of absence, packing my wife into the car, and taking a trip around the U.S. Is it a bad idea to attempt this in winter? I’m Southern Californian born and bred, so my experience with snow driving is limited, but I’ve had some.
Get a map of the U.S., draw a line from wherever you are to Memphis, TN, through Nashville, and on to the Atlantic Ocean. Stay south of that line, and you probably won’t see much snow in November and December, unless you’re in the Rockies somewhere.
If you’re concerned about it, perhaps take a southern route. You are already in the south (SanD).
Where do you want to go? Have you every driven across the great plains? Even if the weather is good, it’s not great fun.
Check out Sante Fe, And maybe Austin.
Coming up to Colorado should not be a problem, but make sure you have good tires.
IMHO, coming across on 70 or 80 in November and December will likely be a non event. In more ways than one. But you might also get quite a bit of winter driving experience twords X-mas.
What are you driving?
Better check that the mountains will be clear of bad weather even in the south. Mountains can have cold weather when surrounding areas are warm indeed. My parents drove through a snowstorm in the mountains in Wyoming. In July. 
I drive a 2009 Prius, with decent tires, and of course I’d carry chains. What about Bryce and Zion in Utah? Out of the question, or still pretty nice in late fall?
I’ve been driving in severe winter conditions for 25 years (and boy, am I tired!), and I don’t drive in the mountains in winter if I can help it. The cross country trip sounds super, but avoid high mountain areas - “some winter driving experience” is not enough. You know they actually close some mountain passes in winter, right?
Zion should definitely be do-able, Bryce might have some snow issues. A roadtrip across the desert southwest can be awesome in the winter, but there’s also a chance of very cold and windy conditions.
May be a non-event. Three years ago, the trip from Chicago through Nebraska in the first week of December took 32 hours instead of the usual 16. The freezing rain all the way from Chicago to the Colorado line did not allow speeds above 45 mph. It was some scary shit.
You may have no problem, you may go sliding off the road at 50 mph.
I would recommend I70, and south, if you really want to do this.
I took a winter trip across Minnesota and the entire width of South Dakota to hike the Badlands and continue onto Mt. Rushmore - solo. I did have the entire Badlands nearly all to myself. I saw more deer than people, and it was a spectacle - like Mars under a foot of snow. It was insane, and I would have frozen to death in my tent in the back-country had I not taken good winter camping precautions and 0 degree rated and field-tested supplies. I also almost died sliding off of a mud-slicked cliff on Saddle Pass Trail. And I again cheated death when I careened off of the highway in Minnesota and just missed other vehicles and steel posts.
So even without camping and hiking, the roads were extraordinarily bad for a good portion of it. I would keep south. Austin, TX, New Orleans, LA, ect.
In late November of 2006, we did it. We started with good weather in Hermosillo, worked our way up through Moab, Utah, on to Mt. Rushmore, a trip through the badlands, on down to Chicago, and then home to Michigan. The only real weather event was snow through some mountain pass west of Denver that delayed us for a couple of hours (i.e., sitting, waiting for jacknifed truck to be moved). Oh, and I had to stop and buy a battery for the truck on the way. What was a great battery in the Sonoran desert didn’t stand up to the cold waking up one morning in Cheyenne.