Open your phone’s voicemail app, then tap (or in some cases, tap and hold) the message you want to save. You should be presented with a list of options; the save option will usually be listed as “save”, “save to phone,” “archive,” or something similar.
AT&T: If you’re an AT&T customer, you have the option of creating your own basic voicemail. You’ll need to press and hold the 1 key down while you’re on your phone. AT&T will then ask you to set up your greeting via voice prompts.

If you’ve already created a custom voicemail on your iPhone, you can tap Play to see how your voicemail sounds. If you’re interested in creating a new message, tap the Custom button. From here, you can start recording your message by tapping Record and tapping Stop whenever you’re finished. Once you’re finished recording, tap on Play to make sure the message comes across as you’d like and hit Save (in the upper right) to save it as your voicemail.
On your iPhone, go to the Phone app and tap the Voicemail icon. 2. Choose a greeting and password and click Set up. 3. Then, confirm the voicemail password you just created. Ideally, passwords will contain no more than seven digits, but at least four. 4. You may choose between default or custom greeting on the next screen. Pre-selected defaults