How to transfer/copy Notes from one iDevice to another without using restore method?

With iOS 9:

Open the Folder containing the Notes you want to move.

Click the Edit button. It is faint. You might not see it right away.

You will see a column of selection circles ⭕️ appear, and also the words Delete All and Move All.

Tap the circles for your target Notes, then Move To ..., and the destination.

All done.

As it turned out, Apple does not provide an option to auto sync/copy previously created notes for Notes app to iCloud in order to sync to other iDevice. From this doc:

"In Notes on your iOS device or Mac, verify that your notes are not stored in a local folder. Example: On My iPhone or On My Mac. If you have notes in these folders, you will need to copy or transfer them to iCloud or IMAP to sync with your other devices."

Well you will need to literately do what what doc said: copy/paste one note at a time to iCloud. If you had previously create notes and once you signed in iCloud account, the Notes app will show two accounts: "On My iPhone (or iPad)" and "iCloud". So you just need to do it copy and paste one note at a time the old passion way. This is no fun.

Its very simple, try Airdrop, open the notes on iphone from which you wish to transfer it, tap on Share icon, tap "AirDrop", and simultaneously enable "AirDrop" from other iphone, choose "Everyone" ,and you are done with it. Only drawback is that we have transfer notes one by one.