Drupal makes it very easy to modify a theme, and still be able to apply updates to that heme as they are released. This is done with a subtheme. Essentially, it is a level on top of an existing theme that changes that theme. Leaving the underlying theme untouched. Any updates to the underlying theme can be easily applied without having to repatch changes.
Generally speaking, a custom template goes into the theme's folder. But what if you want to a custom template for a content type your module has created. By default, Drupal 7 will not search your modules directory. It turns out other people have already solved this for me and I found the answer here.
Add the following code into your mymodule.module file. Don't forget to change the "mymodule" refererences.