Our family lives in Minn, and we've been exactly to what you're planning several times. I can post photos if you like. Rapid City is great. It a fantastic place - give yourself enough days. We've done everything you listed and more. How old are the kid(s)/son - that will matter per hikes.
But key WHEN are you going? Weather will matter, if before end of May. Things you listed can each take at least a half-day. So plan accordingly. Sundance, Hot Springs, Custer, Hill City all have nice places to chillax and eat. When me and my 3 boys did Harney's Peak (bucket list do), my wife and her mom/etc just hung out I think at Hill City and drank G & Ts with appetizers.
**If the weather hits you and it's just raining cats and dogs, aside from getting lost w/ tourists at Deadwood, in your backpocket there is also Jewel Cave or Wind Cave. They are both really cool experiences, where the rain doesn't matter. Book tickets in advance. Setting rain aside:
Day 1: Gotta do Badlands. No brainer. We found it best to enter on the eastern entry, Cactus Flat (where the ICBM museum is) the work the half-loop (called route 240) westward. Best time to go is either sunrise (can't practically unless camping) best y
ou gotta be there for sunset. That's when the light hits the right way and the colors really come through. That's like "ok, so this is where we came west in the first place." So, do a morning different thing, then afternoon/evening Badlands, then back to RC to eat. (See below on Wall Drug, which tourists either love or hate - we're the latter....)
Entering on the east, I think we stopped briefly at the first overlook, on one's left. Then up the road a tad, I recall we walked Door Trail. That's cool. Then back, and kept driving back westward on the loop. There a hundered places to stop, if you want. KEy was timing: when sun set, we were at about Dillon Pass or The Pinnacles, right near the western entry/exit. Just time it for sunset, bring good camera!
En route from RC for morning/noon? Don't know if it's up your alley, an overlooked gem is
Ellsworth Air Force Base's museum. It's free, and if your kids are into big jets etc., you'll love it. This was like 5-6 years ago, B-52s, B-1s, and so much more. Some places are time-traps: once you're in, you're stuck for hours. Not Ellsworth. If you want to hang there for 30 minutes, cool. Two hours, will be fine. Raining - works too, as there's a lot inside. All free; no hassle (back then). Google the museum, easy hit as you are driving east anyway to badlands. If you really want, there's also old ICBM sites off the interstate as you leave Ellsworth to the northern rim entry/Badlands. So yes, there's a intercontinental ballistic missile museum to see. You can dream about nuking Putin, in a missile silo.
Day 2: Do Mt Rushmore/Needles/Custer. See below on avoiding Crazy Horse. I think when we did Mt Rushmore, we matched it with
the Needles Highway and overlooks. Do that - it's cool. You're driving it. Lots of overlooks, photo places, and tunnels carved into the mtns. (again, time of year enters here for spring vs long winter). Cathedral Spires has a hike to consider, but even it you don't want to do it, you can park and see the awesome views. It's on the Needles scenic drive. I think we did the drive first, then Rushmore. But you could do either sequence.
Late afternoon/evening.
You can do Custer, and try to catch sight of a buffalo heard. The donkey's will come to you. It's all driving, unless you intentionally want to park and walk. We drove it. If you do Rushmore first, the do the Needles from west to eastk, you can finish Needles around Custer, and go through it by car. Custer really is a place to camp or to stay at the lodge there. If not, you can see what you want by the drive. If you want a doable hike, the Lovers Leap Trail is a 3 mile ascent, lots of flowers, pretty popular. But it's there if you want. Again, we basically have just driven the park, based from Rapid City.
Day 3 Harney Peak, now Black Elk Peak. (Harney was a general, anti-Indian). Your exercise day. It is the highest place in the north America between the Rockies & the Pyrenees Mountains in Spain. We did it in 2.5-3 hrs roundtrip. Must do - you just can't see this even in VA's Skyline, the Appal. in Tenn/Ga.
Where your going to, is that little cropping of rock on the leftmost side of all that granite. There's shelter there, and the photos are absolute "life bucket list".
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Day 4: Devils Tower and Spearfish. That's a bit of a drive, there and back. You can stop for gas, etc at Sundance. Watch out if yhou overlap with the national motorcycle rally at Sturgis. We stopped at Spearfish canyon and drove through it. There's a couple of places in the canyon drive to do walks, get photos. Th Tower is cool. About an hour of time actually there, as you walk the path around it. You'll see the free climbers scaling the top, or sitting on it. You can whistle the theme and wait for the spaceships too arrive...... Either en route, or on way back, you can stop off at the last key spot,
Deadwood. See below on guidance.
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AVOID:
tourist traps of Wall Drug. You'll see signs 400 miles away. It opposite the South Dakota Badlands (N Dak Badlands cool too.) If you must, like bathroom break, then ok. But just be aware, it's a vast SD tourist trap.
Avoid Crazy Horse. It's a privately run con, and long-line plus big cost to get in. Spend more time at Rushmore, which you can do all day almost. Great facilities, incl. cafeteria. There is short walk up to the mtn base, or you can try longer hikes u that go up the big hill. We only did the base, which is fine. The museum is cool, at the base.
Many SD towns live off of cheap gambling. It's everywhere. So be on guard for that. Still, Deadwood has some things to see - the cemetery on the top of the canyon has Calamity Jane, Will Bill, and others buried there. If you have a car/van, it's steep, but you can drive it - regular paved roads. The view from up there, down into the Deadwood canyon, it something.
