You can disable voicemail by dialing ##002# on your phone. That disables all call forwarding so no calls will go to voicemail. I use Google Voice for voicemail so I disabled it on the cellphone side.
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.

Also, carriers have a limit to how long they'll let you record a voice mail. Test it with your phone and a friend before you trust this method.
When it comes to features and functionality, the Samsung Galaxy phone is outstanding.