I think Mt Rushmore is quite a hike from Yellowstone and the Tetons.... a few years back we flew out to Salt LAke City and drove up to that area- we took approx 9 days just at Yellowstone and Tetons- I have a few suggestions based on what we did- we had 2 teens with us- and we were trying to keep things as cheap as possible- also our goal was to spend time SEEING the park(s) not just visiting each one. We wanted to stay WITHIN the parks, to lessen the driving ,etc and to just BE there.
We drove from SLC up to the Tetons(take the main highway,not the bear lake route to save travel time,lesson learned) and stayed in a VERY basic cabin tent within the Tetons park...it was CHEAP, but it worked- 4 pull down cot/bunks,a concrete floor,and a canvas tent with a woodstove for warmth. (it was mid/late June,and in the 30's at night) the tent was... what it was, the view while eating our breakfast every morning was breathtaking. We loved it. Spent 2-3 days here hiking the simpler trails,(jenny Lake ferry,and trails are great) I LOVE the Tetons!
Then we drove up to Yellowstone for the next week, and stayed 1-2 nights in different places along the main loop road.while sightseeing around that loop. there is SO SO SO much to see in Yellowstone, it is the most amazing place on this planet,I think.
Don't rush it, stop,stay,hike,enjoy the sights. I think we stayed in Old Faithful basic cabins,right next to the geyser area/hotel. (this was our favorite place,it was magical, a wonderful cabin, and morning coffee in the woods with the bison and geysers steaming everywhere was...incredible. This made it easy to spend evening/night walking the boardwalks and listening to ranger talks, this place is beyond belief and nightfall is even cooler.
We also spent a night at Mammoth Springs, it was hot,and interesting, but only for one night,we were glad to move on. Then we stayed at Grand Canyon of the Yellowstone area in a basic cabin/motel room. this area surprised us- I've seen the "real" Grand Canyon too...this one was.....amazing. The colors,the scenery,the walking, the hikes....Just....amazing. Again, surprised us with how wonderful it was.
Each 'area' in Yellowstone has its little 'settlement' with camp stores, some restaurants,and good cell service. I would recommend the entire loop,and mapping out a plan of places you really want to see. the weather was anywhere from snowing (in JUNE) to 95 degrees at midnight,depending on where we stopped....
trip advisor .com is invaluable for planning.....