Notes Synchronization Mess
It took almost 90 minutes with Apple to get a final answer to the problem of updating Notes and syncing them with your iPhone and iPad. The short answer is, you can't, directly.
Preserving Your Notes from AppleMail
To be specific, if you have been using the Notes feature of Apple Mail.app, they do NOT automatically migrate to the new Notes application when you install Mountain Lion, but they do disappear entirely from Mail.
If you are still using Snow Leopard 10.6 or Lion 10.7, you have to prepare in advance to save your notes before running the Mountain Lion 10.8 upgrade. The process is:
Open TextEdit or your favorite word processor.
Open Mail and go to your Notes. Open the first note.
Select All (Command-A) and Copy.
Switch to your Word and create a new document, if you didn't already.
Paste. Then type a couple of returns at the end.
Go back to Mail and open your next note.
Repeat the process, above.
Repeat all until you have gone through all your Notes and pasted them into that Word document. Save as you go along!
(If you are still using AppleWorks, don't! It does not work in Lion or later.)
Notes on iPhone that Don't Sync
I had a bunch of notes on my phone that did not sync via iCloud, even though any new ones I created did sync. The process of rescuing those is even more convoluted, but it works and it's your only option, as I determined during my session with Apple.
On the phone, open Notes and go to the first one you want to rescue. Hold your finger down on a word until the magnifying glass icon appears under your finger. When you let go, you will have a choice between Delete, Select, and Select All. Choose Select All. Then a new popup will appear: Copy. Tap that.
At the top will be a button called Accounts. Tap that. You should see several categories:
From My Mac
Notes (Mobile Me) You may not have this one.
Tap All iCloud. Click the + at the top right of the screen. A new empty note will appear and the keyboard will pop up.
Press and hold in the typing area and you will get the magnifying icon again. Paste will appear. Tap that. The entire note you Copied should appear.
Tap Done. Then tap Notes. Then tap Accounts. Then tap From My Mac and that will take you back to the rest of your notes.
Go to the second note and repeat the process. Keep doing this, note by note, until you have finished the last one.
This seems like a daunting task, but since it takes the same amount of time for each note regardless of how big each one is, it should take you about a minute each; less when you develop a rhythm.
Yes, I agree, this is one of Apple's bigger screwups. The AppleCare techs also agreed.