Thanks for the replies, everyone!
Since no one has done it yet, I’ll list a few things for Cincinnati. It’s not my city, but it’s nearby and I’m a frequent visitor.
*Go to Fountain Square and get a picture of yourself (or -selves) in front of the Tyler Davidson Fountain. My observation is that it’s the official backdrop for “we were here” photos.
*See one of the museums at Union Terminal, or if you don’t want to do that, at least go inside and check out the main rotunda area (or whatever it is called–kind of a big half dome). The parking lot is expensive; park for free on the side street next to it.
*Walk across Cincinnati’s landmark bridge, the Roebling Suspension Bridge. From the bridge on the Ohio side, walk east along the river to around the “Purple People Bridge,” where the riverfront is much nicer, with a couple parks.
*There are more museums, of course. If you go to the art museum, have a look around Eden Park in which it sits and Mt. Adams, an adjacent fashionable neighborhood.
*You can go to the top (the roof, I believe) of the tallest building, Carew Tower, for something like $2.
*See the views from some of the hills surrounding the downtown area. (This is one of my favorite things to do, as I especially enjoy urban “scenic views.”) Devou Park in Covington (KY) is provides the classic view. Also in Kentucky you can find nice views from hills in Newport and Bellevue. In Cincinnati, there are interesting overlooks at Mt. Echo Park, Eden Park, Cincinnati Christian University, some spots in Mt. Adams, and probably places north of downtown that I haven’t personally seen yet.
*See a Reds game if you like baseball. The cheap seats are only $5, I think.
*You absolutely MUST try the chili if you haven’t before. Skyline Chili is ubiquitous and is the place to go for the everyday experience. (4th and Sycamore if you’re downtown, but you’re out of luck at that one on a Sunday.) However, there are many independent places. I hear good things about Camp Washington Chili.
*On your way to or from the things in my OP (Dayton), be sure to see the Giant Jesus along the highway!