How do I set up voicemail greeting on my iPhone 16 Plus?

Open the Phone app, tap Voicemail at the bottom of the screen, tap Greeting at the top right of the screen, tap Custom.

Also see: Set up your voicemail on iPhone - Apple Support

and

Change your voicemail greeting and settings on iPhone - Apple Support

Over and above what FoxFifth has said.. Note: Voicemail and Visual Voicemail are services provided by select carriers in certain countries or regions. Availability may vary depending on your mobile plan and location.

If your carrier supports Live Voicemail, you can set it up and use it by following the detailed instructions provided in the articles shared by FoxFifth