The first such ice cream place in my memory was a non-DQ one at the corner of East Main St. and Yearling Rd. in Whitehall, a then-new suburb of Columbus. We had moved to a new house (costing $12k) just off of Yearling when I was around five, and visits to that ice cream jernt were a treat.
The Dairy Queens and other fast food places I most love to find are those that still have their original building and signage. There's not a lot of that left, but I loved finding a Dairy Queen like that on Route 4 (I think it is) a few years ago in Cincinnati, a little south of Springdale. I've seen one or two others in Ohio, too, somewhere...