Google Photos/Backup has an automatic backup feature. My initial idea of saving my app's photos to the Gallery does not work since I cannot access them by code. What is the proper way of saving my photos/files so that it will be backed up by Google Photos/Backup?
Google Photos will not backup your images if they aren't considered to be in the gallery.
There is a built-in backup feature which is enabled, unless you explicitly set android:allowBackup to False in the manifest editor.
It should backup your app files automatically. Though I don't think that you have any control over its exact behavior.