Time for Some Great Hikes
As I mentioned earlier, your major constraint is time. While 10 days may seem like a lot of time, once you factor in a day's flying time each way, and a day or so to drive between Denver and Rapid City each way, you're left with only six days or so for everything else. Trying to include a third state would eat into those available days without really getting you too much else than being able to say you'd been to a third state. My first thought was to do Denver, Rapid City, Yellowstone and back to Denver. But between the extra driving time and the crowds that can be expected at Yellowstone this year, as everyone starts to travel again, would make that more work than play.
So, my first suggestion would be to fly in and out of Denver as the most cost-effective option and then just do an out and back trip, or a narrow loop, to the Rapid City area. As I said there are lots of great options there including Mount Rushmore, the Badlands, Wind Cave, Jewel Cave, the Crazy Horse Memorial, Devils Tower and Thunder Basin National Grasslands. And of course, Rocky Mountain NP near Denver. There should be good hikes in pretty much all of those sites.
AZbuck
Hike recommendation: Wild Basin Trailhead,
As with many national park sites Timed Entry Passes are required at Rocky Mountain National Park this year. Their website states they will be needed from May 27 through October 10, 2022. You can find out the details under the icon Info, then Plan Your Visit.
The reservation is made through Recreation.gov. I would recommend determining a date as soon as possible and securing your permit now. Some parks have a small percentage of next day permits, but that is risky.
Hike recommendation
Wild Basin Trailhead, east edge of park along CO 7 to Ouzel Falls. I can still ‘hear’ the rushing water along the creek heading up to the falls. The website has a good trail map showing parking and mileage.