Saving notes to icloud

Ensure you've followed the steps to turn on iCloud for Notes - Set up iCloud for Notes - Apple Support

1. On your iPhone, iPad, or iPod touch, go to Settings > [your name] > iCloud.
2. Turn on Notes.
3. To view notes stored in iCloud, open the Notes app. In the Folder list, select All iCloud or a folder in the iCloud section.

Alternatively, if you have a Mac, you can export notes as a PDF - Import and export notes on Mac - Apple Support

1. In the Notes app on your Mac, click the note you want to export.
2. Choose File > Export as PDF.

If you're getting a new device, though, Notes will be transferred with Quick Start, if using, or by setting it up from a backup (either iCloud or PC). You can read more about that here - Transfer data from your previous iOS device to your new iPhone - Apple Support

On your old iPhone the notes application box should always be turned as on Set up iCloud on all your devices - Apple Support so that whenever you create notes from the device it will sync with iCloud , when new iPhone is purchased and set up follow the article Set up your iPhone, iPad, or iPod touch - Apple Support the notes data from old iPhone will sync to new iPhone ( make sure same Apple ID and password will be used on both device and they should be on the same network .

If notes box was not checked in old iPhone , then its impossible using old iCloud back up to transfer notes in new iPhone .

What does iCloud back up? - Apple Support

Your iPhone, iPad, and iPod touch backups only include information and settings stored on your device. They do not include information already stored in iCloud such as Contacts, Calendars, Bookmarks, Notes, Reminders, Voice Memos4, Messages in iCloud, iCloud Photos, and shared photos. Some information is not included in an iCloud backup but can be added to iCloud and shared across multiple devices like Mail, Health data, call history, and files you store in iCloud Drive.

Its essential notes should sync to iCloud so that data with it will save on iCloud server .

When you turn iCloud for Notes on, it will automatically save all notes- current and future- in iCloud for you.