Open the Phone app
Tap the menu icon or More options
Select Settings
Tap Voicemail
Tap Voicemail greeting or Greeting
Choose Record a new greeting or Custom greeting
Tap Record
Speak your new voicemail message
Tap Stop when finished
Tap Save
Set the new greeting as active
If prompted, call your voicemail and follow the voice prompts to change the greeting
If your carrier uses a separate voicemail app, open that app and change the greeting there