A couple of thoughts for your consideration.
We like to drive route 89 when going from SLC to Grand Teton NP. Between Logan and Garden City 89 is the
Logan Canyon Scenic Byway. After Garden City we stay on 89 all the way up to Jackson. It's a beautiful drive with one 6% down grade just before Garden City. Use lower gears with the RV going down to save the brakes. If you decide to use I15 up to Idaho Falls, take route 26 to Alpine then 89 up to Jackson. Do not use route 22 from Victor over the Teton Pass to Jackson. 10% grade, not recommended for a first time RV'er.
Gros Ventre is our favorite campground for the Tetons. When driving Gros Ventre road between 89 and the campground, keep your eyes open for Moose grazing along the river.
We have had snow showers as late as mid-June on several past trips to the Tetons and Yellowstone. Bring some warm clothes so you can layer up.
2003 Winnebago Minnie 24F - Ford E-450🙂