We did Hwy 14 over the Bighorns this past summer, pulling our 26.5 fifth wheel. It's a many mile 6% grade up, and the same down. Lots of hairpin turns with a max speed of 20 mph. Not frightening, but slow, demands good braking skills, and attention to the road. I'd call it very do-able and a good introduction to mountain driving! Do NOT take 14A from where it splits from 14 in the Bighorns. That one is reportedly a real hair raiser, even if not towing.
Hwy 16 is an alternative - haven't driven it lately, but seem to remember it as about the same as 14. Many forest service campgrounds in the Bighorns, as well as off road camping.
There is a more southerly route, going to the Tetons first, by way of I80 through Nebraska and Southern Wyoming that looks like it may be pretty fast. We haven't done that in many years, so can't give much of an opinion about it. It will be very flat country, though, all the way from home until you actually reach the Tetons. And even then, the road up to Yellowstone is a pretty easy drive. Might be the best route if you plan on putting the hammer down and driving well into the night!
2012 Ford F-150 Eco, Max/Max
Jayco 26.5 RLS, Reese Sidewinder