Well, Bruce, your child's doctor (pediatrician) could check your feet, but you'd be much better off at the podiatrist.
I used to do a lot of folk dancing on a wooden gym floor and asked one of the dancers who led the group about her selection--she wore a very comfortable flat soled boot that didn't quite reach mid-calf. She had them custom made for dancing. The dancing we did were a lot of Scandinavian and Eastern European dances (a lot of running and fancy footwork on the move). Sneakers/running/athletic shoes would tend to "catch" on the floor when you want to travel or slide your feet. I'd find something flexible with some cushion but not a heavy or thick sole. And probably have to switch out shoes every few months because they won't last as long as sneakers.