I know that SZ0 makes vertices straightened on the z-axis, but I'm wondering if there's a way to make vertices into a complete circle on the z-axis with a hotkey?
There's an add-on called "mesh tiny cad" which is still included in the standard installation of Blender (2.79 and 2.8+).
- Pick any three vertices in Edit Mode
- RMB -> TinyCad -> CCEN,
You'll see a circle of verts which you can adjust (radius, num of new verts)
This add-on/operator doesn't care about the orientation of your three vertices, it's not limited to X,Y or Z axis. It can be an arbitrary plane.
small documentation page for the add-on: https://zeffii.github.io/mesh_tiny_cad/
Loop Tools. Click the image to enlarger.
You can install this addon via User Preferences. In the later version of Blender the addon appears with the context menu. (Click with the other mouse button). Now that you know the name of the addon you can see videos about how to use it on the famous video website or exactly here by searching BSE.
I have used some of the other techniques in past and present, presented in the other answers. They work well. Loop Tools has many interesting features which go beyond the scope of your question for your future endeavors. There are always 33 ways to do something in Blender.
Some images may be from an earlier version of Blender.