Looking to stop at several places along the way

Krissy has some great suggestions. I will add a great stop for 5 Star dining between Bozeman and Billings there is a small town called Big Timber right on Interstate 90 which has a historic hotel downtown that, to me, is one of the best places to eat for dinner. I would continue on I 90 and if needed a stop to sleep in Buffalo (Wyoming) at the very nice Hampton Inn & Suites Buffalo and a downtown trip to eat at their old hotel. Then off to the Devils Tower and the Black Hills. The Badlands National Park of South Dakota are very similar to Theodore Roosevelt National Park in North Dakota so you would not have to see both. I would consider going through SD rather than heading north to ND and stop in Sioux Falls and Falls Park. Then east to Worthington (Minnesota) and take highway 60 north to 169 into the Minneapolis area.

I agree with Devils Tower National Monument, it's a nice day visit but if you want the campground was very nice $20. The park offers alot of programs. I think there are a lot better places to spend your time than Mt Rushmore. Theodore Roosevelt National Parkwas quiet and beautiful. Hwy 87Needles Highway in Custer State Parktown of Custer.

A couple of comments. Sun Valley is a nice days drive from California and a great base for a couple of days any time of year. Same comment for Jackson Hole. A special stop is Cody Wyoming and the museum there. Check it out.  

Beyond Mt Rushmore South Dakota and Nebraska are flat and boring. Sorry but it is true.

My answer is based on a road trip from Portland to Rapid City, but I think some of the same suggestions apply.

Boise seems to be a bit too far west to be efficient for your route, but Craters of the Moon National Monument is interesting and has some cool caves you can explore. We took Route 20 and stayed a night in Idaho Falls before heading up through Island Park (Idaho) on our way to West Yellowstone and then Yellowstone National Park proper. I second the previous suggestions about Grand Teton National Park; it is beautiful and worth the time. We enjoyed stopping by Jackson Hole.

We spent a lovely, restful night at the El Western Cabins & Lodges in Ennis, MT and visited Yellowstone the day before and after. We found the The LARK in Bozeman, MT to be a fun and quirky stay and appreciated the small town feel. We found great food in Bozeman: Blackbird and Wild Crumb.

Our route took us east on I-90 and then I-94, but I have to say that we didn't find anything very remarkable in eastern Montana (with apologies to anyone who feels otherwise; it might have been the time of year). We wanted to visit North Dakota and did so via a stop in Medora (North Dakota). It was all but deserted, and I'm sure that was because of the time of year. During a different season, it would have been much more lively. We found Theodore Roosevelt National Park beautiful in a desolate way and were grateful to experience the wild landscape with no one else around. 

If you're going to Minneapolis you might head east at that point. If you have the time and inclination, you could turn south through Deadwood (our highlight was Mount Moriah Cemetery). We stayed in Rapid City and made trips to Mt Rushmore, Crazy Horse Memorial and Jewel Cave National Monument. Badlands National Park is close too.

We spent about 7 days going from Craters of the Moon to Rapid City:

1. Craters of the Moon to Idaho Falls
2. Idaho Falls to Jackson (Grand Tetons)
3. Jackson, West Yellowstone/Yellowstone, Ennis
4. Ennis, Yellowstone (again), Bozeman
5. Bozeman to Medora
6. Medora to Rapid City
7. Side trips from Rapid City

Due to the weather and time of year (April) when we made this trip, some roads were closed, so it wasn't the most efficient. That made it slightly challenging to plan, but we loved not being in crowds and experiencing this part of the country, especially the National Monuments and Parks, mostly to ourselves. There were also lots of baby animals around.

Have a lovely trip! I hope this helps.

 I agree with Fritzi San Francisco to Elko ( might also stop in Reno) to Twin Falls and then into Grand Teton National Park and on to Yellowstone National Park

From there exit the park through Cody you might go up to Devils Tower before you go through the Black Hills National Forest to Mt Rushmore to the Badlands National Park  on the way to Minneapolis

We drove from San Francisco to Yellowstone National Park; broke the trip in Elko, NV. Travelled briefly through Idaho and down Teton Pass (magnificent) to Jackson (Wyoming) and spent time at the Tetons and Yellowstone. Can’t route you the rest of the way as then we came back home to SF.