“Passwords, messages and notes are not syncing with icloud: Hello, after switching to a new iPhone and logging into iCloud, my passwords appear in the Passwords app, but they don't show up when I try to use them in other apps. Also, Notes and Messages aren't syncing.”
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iCloud Time to Sync:
A. Wait it out. It may take some time to sync.
B. Third-Party Apps Requiring a Passkey.
What do you mean by “other apps”? If they are third-party apps, then you may need to create a new login password for them, due to this being a new device for these apps. Such an app would be for email accounts.
C. How to Transfer These:
As for setting this up, this is the site to use: Use Quick Start to Transfer Data to a New iPhone or iPad - Apple Support