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.
Record your greeting. Under Voice mail, select Greetings. Select the type of greeting you want to record. Select Call me to play or record the selected greeting. Enter your phone number and select Dial. Follow the instructions in the recorded message to record your greeting. When finished recording, click Save.

Insurance agents have to deal with large numbers of clients almost all through the year. So, they are extremely busy and cannot take calls of clients. For them resorting to voicemail messages become inevitable. Here is one instance of VOIP phone service for businesses like this:
In the T-Mobile Visual Voicemail app, for example, tap the three-dot menu at the top right and then choose "Settings." In the Greetings & Pin section, you can tap the default greeting and record a new one.