I would suggest taking highway 16. This is a more gradual climb up and down the mountain. You are able to travel at a faster speed and there are not as many "switchbacks".
When checking on a map program it says that it takes less time to go on hwy 14 but I do not think that is the case because the road has a lot of curves.
Both routes have great scenery.
Watch out for deer, elk, antelope while driving. This is especially true early in the morning and at night.
I had been assuming that you would be traveling south west first and then heading north. If you are flexible you might want to fly into Denver spend a day or two there and check out the weather forcast. If it looks nice maybe start your trip north to South Dakota and then come around to Yellowstone and then head south.
It sounds like you will hit a lot of National Parks - if you do not have one get the NP pass.
$$ saving tip. If you do not have a lot of hotel points - try some of the mom and pop motels to save some money. Some of them in SD, WY, CO, UT are not to bad and can be inexpensive. It has been three years since I travelled in that area but I can remember seeing signs for motels between $30-$60. Of course ask them to see the rooms first to make sure they are clean.
I have a national access pass due to being deaf, so I'll be able to get into all the national parks for free. Thanks for the Route 16 suggestion too, I've since updated the itinerary to pass through Worland and Buffalo on the way to Mt Rushmore.
