Rent the car, I have done this twice from Vegas and highly recommend it. Be sure to stop at a grocer

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As time gets closer go to www.weather.com and put Williams, AZ into the search box and then choose the 10 day forecast to see if weather is going to be a problem. You might also put Las Vegas into the search box too. As long as it is not snowing when you plan to go you'll be fine.
And when I say bad weather, I mean a winter storm has either just occurred or is about to hit that day. There will be snow on the ground at the canyon but generally roads are good to drive unless there's been a recent walt disney world hotels storm or significant snowfall.
Rent the car, I have done this twice from Vegas and highly recommend it. Be sure to stop at a grocery or the dreaded Walmart for a small Styrofoam cooler for water, good to have. Leave early in the morning, it is breath taking!
We rented a car and drove from Vegas to the Grand Canyon in December. I would definitely recommend that. Most Vegas hotels offer free valet parking so no expense walt disney world hotels there. The drive was a very quick and beautiful one. We spent the night literally right on the Canyon's edge at Kachina Lodge. (Call, they get cancellations.) I feel this is the best bang for your buck. It is amazing to be able to look out over the Canyon in the quiet and dark of night, a sky full of stars. You will be able to drive the park at your own pace, pulling over for incredible views wherever you please. On the drive back we stopped at the Hoover Dam and Lake Mead. I wasn't in awe of the lake, personally, but did enjoy viewing the Dam from the Pat Tillman Bridge.
However,what concerned us most was the prospect of driving in road conditions that we have never driven on before ie ice/ snow ( remembering it will be February 3) while also driving on "the other side of the road".
Just because it's February 3rd doesn't mean there will be ice/snow on the road. Unless there's just been a major storm, the roads will be clear. walt disney world hotels Really the biggest chance of seeing any ice/snow on the road would be the last hour of your trip from Williams walt disney world hotels to the Grand Canyon. Prior to that, you're travelling through the desert or on a major interstate so little chance of any ice/snow unless there's been a recent storm (in the past day or so).
I live in a cold climate, far colder than the Grand Canyon but our major roads are all typically clear and dry within a couple days following a storm. Occasionally it will take a bit longer to get them completely clear if it's been a massive storm, but a typical snowfall is cleared off and dealt with pretty quickly.
Lavendar we drove to the canyon from Vegas in October and as Aussies we found driving on the wrong side of the road fine. The only time we had to really think hard was turning as we turn loosly when we turn right and tight when we turn left. So we had to keep saying.... right and tight, left and loose everytime we turned. It felt strange for a while but we soon settled in to it and driving mostly on highway means its easy. Of course, we didnt have snow or bad weather but the drive really was easy and the highway is good to drive on. We hired a GPS to make sure we didnt get lost and to help us get in and out of Vegas. We even detoured and did a bit of the old Route 66 and stopped for lunch at Seligman on the way and had the best club sandwich we have had. We found the drivers all kept to the right and didnt clog up the lane for others to overtake (unlike over here sometimes)