We are leaving from the Best Western in West Yellowstone.
Jess
Why I asked is because it will determine your route. (US 20 to I-15) is the fastest route from West Yellowstone) I have driven about this same route to LV many times. You can average 75mph, the common speed limit but in Southern Utah it's 85 for a ways, and through SLC slower but traffic will move.
As you travel south from West Yellowstone, about 5 miles south of Rexberg, ID on the right side of the divided highway US20 is Bear World.
It's primarily a drive though zoo. You can see some odd animals. Some non native animals. And of course lots of bears and some wolfs. It's ok if you haven't seen a bear before. But the ones at bear world are fat and lazy and not nearly in shape compared to bears in the wild. It also has a couple carnival rides too.
Under 5 miles off the freeway in McCammon, ID, on US30 is Lava Hotsprings, ID. It has a big pool with water slides, a couple soak pools, all different businesses. And there are a couple companies renting tubes to go down the river that runs through town. It is really wild rafting and I would get a double tub and hold onto the 4yo. The creek isn't big or terrible deep, but swift has a couple waterfalls and exciting rapids. It's very popular place on summer weekends. I've been there for many family reunions.
I also like the idea of vising Temple Square in SLC.
I would do this drive in one day, as I have many times. You gain an hour so with a 10 hour drive plus an hours for gas, bathroom and lunch if you leave at 8am you could arrive at 6pm really not bad.
But if you must split it, don't drive through SLC at rush hour. It is thick traffic for 100 miles. Almost all of the population of Utah lives along that 100 miles of freeway. No kidding. Go through SLC at night or midday. Not 4pm to 7pm, and not 6 am to 9am.