From SLC there are two good ways to get to Yellowstone. The direct route, I-15 to US 20, is a bit less scenic for about 1/3 of the drive, but should take about 6-7 hours at about 60 mph. That will take you right to West Yellowstone, which is a good base camp for visiting Yellowstone (allow a minimum of 3-4 full days for Yellowstone & Grand Teton combined). The longer route, more scenic and definitely more mountainous, we have not taken for many years, but goes up through Logan and Bear Lake to Star Valley, then from Alpine WY up through Jackson to GTNP. That will be a lot of hours, and you won't be to Yellowstone yet; you'd want to spend a day or two camping in the Jackson-GTNP area before going up into Yellowstone and spending another 2-3 (full) days there.
