I've seen so many questions asked about it, and I found a proper solution today while working. See below. I searched around, and everyone's answer has been either 'copy all of your notes manually' or use a '3rd party solution'. This requires neither.
All devices need the New Notes upgrade to be compatible with this.
Update to New Notes on each device.
Open folders view and select Local Notes or On My Phone.
Tap on Edit in the upper right corner and tap on Move All in the lower left corner.
You should see this:
'Notes 0 >' [or not including the local notes]
- Tap on Notes
They will empty out of the local notes folder and move into iCloud.
Again, you won't see any of these notes until you upgrade to New Notes.
Pro-Tip: This also works for migrating Gmail notes, etc.