There are all kinds of routes to take. But my preferred route from Provo is:
Provo to Heber, Heber to Evanston, Evanston to Cokeville through Woodruff, Cokeville to Afton (Star Valley), Star Valley through Alpine to Jackson. This trip can be accomplished in under five hours with good conditions and minimal traffic. Typically fuel is cheaper in Evanston, roads, travel faster, are flatter and reasonably patrolled.
Other routes include going up I 15 through Pocatello or Idaho Falls and cutting across going over the Teton Pass or through Tin cup. Those are neither safer nor shorter, but there are people that insist on taking them. I’ve frequently had this debate.
One of my favorite routes for summertime is through Monte Cristo pass. It works pretty good if you are coming from anywhere north of Salt Lake but it definitely adds time. In the fall it is a gorgeous drive.