Not sure I can add much to the advice in this thread but I thought I'd share my experience. About 12 years ago, I decided to take a couple of weeks and drive around the west coast parks. Left LA and was in Jackson Hole the 2nd night. Jackson is really nice but definitely a tourist town. Prices are up there. In the summer you can still ride the Gondola to the top of the mountain and there are beautiful views. Drove through the Grand Tetons to Yellowstone and spent three days there. There are several different areas but waterfall is great. My favorite thing to see was the bubbling hot springs and terraces. I spent hours taking photos. Surreal and gorgeous. Old faithful is cool once but didn't need to spend a lot of time there. But the park is really amazing and there are things to see for everyone - wildlife, geysers, waterfalls, forests, more wildlife, and the most amazing crystal structures that look other worldly that you will ever see.
For the record, I think you'll get the best flights and car rental rates flying into SLC and driving from there.
If you have time, driving from Yellowstone to Glacier National Park in Montana is well worth it. It's about 6 hours. Glacier is just amazing - particularly the Road to the Sun if you can handle steep drops while 2 feet from your wheels when you drive.
I went from there to Little Bighorn which I would recommend skipping. Very depressing. A mound in the middle of a field with lots of flags where people died.
If you have enough time on the trip, you can drive from Glacier to Mt Rushmore in about 12 hours. Mt Rushmore isn't worth more than an hour or so, but you can see the Badlands which were cool and Sturgis in the same day. Both worth while.
Some of the best parts of the trip was driving between the different parks. Really cool landscape and geology and when they say big sky, you don't understand until you drive across Montana.
Hope this helped.