I added a new folder in Notes but I don't know how to transfer new or existing notes into the folder.
Tap Edit, select the notes you wish to move to a folder, choose Move To… and pick the destination.
- Force touch (if not capable, see next sentence) on the note, swipe up, and select "move". If your phone doesn't support force touch, swipe left on the note and select "move".
- If moving more than one note, press edit, select the notes and select "move to...".
- If either of these don't work for you (need to have the updated version of notes enabled for above options), you can always select all and cut the note, then create a new one in the desired folder and paste.