Flattest, warmest route to Las Vegas in January from NC

I40 has signs that flash for when the snow or conditions get to bad. Should tell you something.
You go over many foothills and mountains on it also.
Bottom line if you want flat is I10.
You do go over some hilly areas as you near Vegas but they are not bad. The only other thing in your winter driving is wind , you are going against it all rgw way. Gas mileage sucks at times.
Have fun
We stay at Arizona Charlies RV park, reasonable and all pull through sites.

If you want to avoid winter weather in January - stay off I-40. I-10 runs close to the border so the sooner you can plot a course to it, the better off you'll be.

I had a friend in Albuquerque, NM (on I-40) I was going to pay a visit years ago just after Christmas. When I told him when I'd be coming through he recommended I'd be better off on I-10 that time of year. I took his advice. We spent the night in Las Cruces (on I-10) and saw on the New Mexico weather report that I-40 was closed at Albuquerque due to a blizzard that was expected to keep the highway closed for 3 days.

Take I-10 to Phoenix and then work your way north up to Las Vegas over Arizona Route 93 (beautiful scenery!). Avoid Flagstaff because it's altitude is 7000 and prone to snow there too.

If you've got a smart phone, Google Maps and WAZE are too very good free apps you can use for guiding you along your route. Just put in where you want to go on the next leg of your trip (like for gas, to eat or spend the night) and those apps will get you there.

If you want to use paper maps, they're still available at the welcome centers on the interstates.

X2 been there. also been in a two day ice storm in texas. watch the weather way ahead of you. couple days ahead and plan for it.for just a quick overnighter a cracker barrel worked great.

I-40 all the way.
You can't avoid hills, might as well take the gentlest ones.
It's winter so why risk some backroute. The freeways are usually the first cleared.