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 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.

Google Voice is a cloud-based communication network which is designed for business of all sizes as well as for personal use. It provides you with a phone number using which you can make calls, text messages, and can send voicemails on both smartphones and computers.
Rehearse your greeting a few times before you press record. Plan your pauses and select natural places to take a breath. If you are recording directly into your phone don't hold it to your ear like you are talking on the phone. This can produce a muffled tone. Hold the phone out in front of you a few inches from your mouth for the clearest recording. This may require some trial and error. So playback your greeting and make necessary adjustments on your re-record.