If your landline doesn’t have a way to save messages already built in, the best way to save that gorgeous message is with a recorder. But before you rush out to get it, don’t forget that most smartphones have a built in recorder. Try that first and see if it will pick up her voice loud and strong. It should save the message as an MP3 automatically, that you can send (via email or text) directly to your computer for safe keeping. If the quality is not great, then you can borrow a recorder or purchase one. We like the Olympus brand ourselves, because they save in MP3 and can upload to a computer wirelessly. We have the vtech phone/answering system indicated above. The outgoing message/greeting is of my young daughter’s voice and we would like to save it somehow. Am wondering if you have a process or can suggest how an outgoing/greeting message can be downloaded for safekeeping.
Set up Visual Voicemail. If your carrier supports Visual Voicemail, you can set it up with these steps: Go to the Phone app, then tap the Voicemail tab. Tap Set Up Now. Create a voicemail password, then tap Done. Enter your password again to confirm it, then tap Done. Select Custom or Default. If you choose Custom, you can record a new greeting.

If you haven’t set up any voicemail on your android, you can follow this step by step tutorial:
You can view and manage the greeting in Greeting Management.