I have moved every couple of years for my spouse’s current line of work, which he’s been doing for almost 11 years, and during that time, have lived in 7 different cities (6 states). I just moved to my current location less than a month ago, so I’ll highlight the state I’ve lived in the longest: KANSAS!
Picking 3 must visit cities, I’ll choose Topeka, Lawrence, and Overland Park, KS. Of course, once you’re in the Kansas City area, there are also a lot of the attractions on the Missouri side so you can cover two states in one stop! And there are lots of other interesting attractions in cities like Atchison, Manhattan, and Wichita, to name a few. And the Eisenhower Presidential Library is in Abilene!
Topeka is the state capitol and I would visit the capitol building (tours are offered, including going up in the dome if you are okay with heights) and the Brown vs the Board of Education National Park. For an outdoor activity, Old Prairie Town is fun, if visiting Tuesday - Saturday from 10 - 4 or Sunday from noon - 4, go to the old time soda fountain and order a Green River!
Lawrence is the home to the University of Kansas and besides exploring the campus, I would visit the Booth Family Hall of Athletics, where you can see Naismith’s Original Rules of “Basket Ball”, and the Dole Institute of Politics, then I would wander up and down Massachusetts Street to enjoy the shops and restaurants.
In Overland Park, I would visit the Johnson County Museum, the Deanna Rose Children’s Farmstead, and the Museum at Prairiefire. And of course, have some barbeque!
I tend to prefer museums and educational sites in my sight-seeing and of course this list reflects that, but there are lots of outdoor attractions if that’s your preference.