Thank you David and Brenden for the help.
As both of you mentioned, you cannot stage module classes and that appears to be correct as far as I can tell. You can stage a module and you can add code (using Brenden's links, which I did do and it does work) to stage class data.
To export the module classes to another site I went to the "Sites" app within Kentico, then "Export", then "Do not preselect any objects". I then unchecked the "Export files" checkbox and unchecked the "Export tasks" checkbox because I only needed the module class objects. I then found the custom module under "Development" then "Modules" and selected that checkbox only. That created the export zip for me. I imported the zip file into the next site and the classes were added correctly. After importing the module classes I was able to sync the module class data just fine.