How do I move existing notes from my iCloud to another folder (gmail, etc)?
Your notes are backed up in iCloud. Go to iCloud.com and you should be able to see them all there.
If you are more comfortable storing them in Google, you can: 


Enable notes under accounts (under your Google account) on both devices
Cut and paste the actual text in your iCloud notes into new notes within Google on the notes app. You can’t just drag them over unfortunately.