First check that all your settings are correct, that notes syncing (mail and notes on a computer) is checked on all devices (system preferences > iCloud on a mac and settings > iCloud on a iPhone, iPad or iPod).
Make sure the notes you are adding are added to your 'iCloud' account and not an 'On My Mac', 'On My Phone' or other non iCloud account (you can do this by checking in accounts on an iOS device, or the left side panel in mail on a computer), non iCloud notes will not sync.
If you are sure that everything is set up correctly and your notes are in the iCloud group, you might try unchecking notes syncing (mail and notes on a computer) in the iCloud settings, restarting your device and then re-enabling notes syncing.
The me address you have been asked to create is merely an alternative address for your Apple ID. It's the address you can use with your iCloud mail, iCloud mail will not work with other mail accounts.