I just installed Drupal 8.1.x and used the Minimal installation profile.
I added a new content type, but how do I add additional fields? I don't see a button anywhere.
The minimal profile lives up to its name - the vast majority of modules aren't installed when using it.
This include the core Field UI module, which provides the UI you're missing. Just enable that module and you should see a "Manage fields" link as the default option in the table from your screenshot.