I will share a route which I have taken out of Jackson. This is not the shortest or quickest route but I have towed our 34' fifth-wheel over most of it.
It is basically to follow US 191 all the way from Jackson to Moab. This will drive through the delightful town of Pinedale, WY. Which is surely worth a stop. As I recall we stopped there for lunch in a local restaurant. We had no trouble finding a place to park our rig.
You will pass thorough the Flaming George National Recreation Area. Worth an overnight and maybe an extra day if you can possibly include that. We found it an unexpectedly scenic area. We had driven though it before, but staying there a day really allowed us to enjoy it.
You will pass through Vernal, UT and a slight side trip (20 miles) would take you out to Dinosaur National Monument. My wife especially enjoyed that visit, but it would again take a bit of time.
I will mention that we continued from the National Monument on US 40 and CO 64 to Rangely, CO then CO 139 over Douglas Pass down to I 70 West over to pick up US 191 near Moab. But I considered Douglas Pass to be a serious mountain drive with a fifth-wheel. We did not have a minute's trouble and I would do it again, but I cannot recommend it unless you have considerable mountain towing experience. (It might be much improved, our trip was several years ago.)
So that brings me back to the simple "191 all the way."