I started my search by wanting to know how I could share to other apps in iOS. I discovered that two important ways are
UIActivityViewControllerUIDocumentInteractionController
These and other methods are compared in this SO answer.
Often when I am learning a new concept I like to see a basic example to get me started. Once I get something basic set up I can modify it how I like later.
There are many SO questions related to UIActivityViewController, but I couldn't find any that were just asking for a simple example. Since I just learned how to do this, I will provide my own answer below. Feel free to add a better one (or an Objective-C version).