Plenty to see & do around SLC. There are of course the sites downtown - Temple Square, the Gateway (outdoor shopping area), Beehive House & other historic sites, the Family History center, etc. This is The Place State Park and the zoo are up not too far from the University of Utah. Thanksgiving Point is quite a draw as well. When we are there to camp, we stay at the SLC KOA; convenient location and way nicer than you'd expect for a near-downtown CG.
Farther north, near Layton, is Antelope Island State Park, reached by a causeway out into the lake. The CG there is open & sunny and can be hot; no hookups. Farther south, Provo has several highly-rated CGs and a state park (with hookups) - we haven't been there yet, but those could be good options.
About an hour north of SLC, Brigham City has 2 private CGs, a KOA and another one, decent enough for a night or two, plus another state park (Willard Bay), which has a CG with hookups. From there, it's about an hour drive west to the Golden Spike National Historic Site, worth visiting if you haven't yet been there. Each day during the season they take the trains out (2 steam locomotives, built like the original ones which met at that point where the RRs were completed & came together)and you get to see them up close.
If you're looking for the lowest, easiest way to YNP, take I-15 up to Idaho Falls, then northeast on US 20. Good roads, 4 lanes up until close to Ashton, and the climb up Ashton Hill isn't bad. If you're uncomfortable with mountain driving, don't take Beartooth - and I don't recommend doing it while towing!
