How do I recover notes and download to new iPhone?

• If your notes were synced with an email account (gmail, yahoo, outlook, etc) enter that email account and they should be there.
• If your notes were synced to iCloud with the same Apple ID you'll use on the new iPhone, just open your Apple ID and they should repopulate to your iPhone
• If your notes were on your old iPhone only, they can be recoverable from a backup you made before switching to the new device (be aware that the restore from backup process overwrite all existing data on your new device, unless you use a 3rd party app, i.e. iMazing or iphonebackupextractor, which let you restore only specific data). Obviously, No backup = No recovery.

Regards

Giulio

If you're missing notes on your iPhone, iPad or iPod touch – Apple Support (AU)

If the notes were deleted, and they are not in Recently Deleted, they are gone.