This is a short article, but we do get asked often how to integrate with the built-in Apple Notes app. Unfortunately, at this time, Apple does not provide ways to integrate with Notes that would allow more direct, interaction-free integration with third party apps. Currently, content can be sent to Apple Notes via its Share extension.
Use the Share action, which comes in the default “Basic” action group and select the “Notes” app. From this share extension, the draft text can be added to Notes as a new note, or appended to an existing note.
If you share often, it might also be useful to install the Share (selection) action, which sends only the current text selection, not the entire draft.
You can, of course, always copy and paste from Drafts into a note in Notes as well.
Hopefully Apple will often more robust ways to integrate in the future. (Hint, hint any Apple people reading this…rdar 13378199, closed as duplicate in 2013 )