Congratulations on your new iPhone! I understand that you created a backup of your notes, and now you're trying to get those notes to your new iPhone. I'm happy to provide some help for this.
If you created a backup of your entire iPhone, you'll want to restore that backup to your new iPhone:
Restore your iPhone, iPad, or iPod touch from a backup - Apple Support
If you were using iCloud to sync your notes, make sure that you've set up iCloud on your new iPhone:
Use Notes on your iPhone, iPad, and iPod touch - Apple Support
Set up Notes with iCloud
You can use iCloud to update your notes across all of your devices. To set up Notes with iCloud, go to Settings > [your name] > iCloud, then turn on Notes. If you're using iOS 10.2 or earlier, go to Settings > iCloud. You’ll see your notes on all of your Apple devices that you’re signed in to with the same Apple ID.
Thanks for using the Apple Support Communities. Cheers!