Open your Phone app. Tap Voicemail. Select Greeting. By default, your voicemail will be setup with your carrier's generic greeting. To record a custom greeting, select Custom. Tap Record to begin recording your voicemail greeting. When you finish recording, tap Stop. To make sure you're happy with your new greeting, tap Play.
Ive been saving my grandmothers voicemails for almost 8 years now and when i went to upgrade my phone, i almost fainted when the rep told me they were going to be erased. Thanksfully he put us in touch with a company called http://www.voicemailsforever.com and they ended up much cheaper than the one you suggested above. i think your readers would benefit because they do it for so much less.

Great question! Although I don’t know if there is a specific function for this on your particular phone, the best way to save the message is the simplest. Turn up the sound as high as you can and record it onto a digital recorder (like you would use for a lecture), a cassette recorder or your cell phone if you have a voice recorder on it. Then save it as an MP3 on to your computer for safe keeping. You can also Google your vtech model number to see if you can find an owner’s manual. They might have a way to save a recording in there. Probably not, but it’s worth a shot.
Visual Voicemail makes it simple for you to keep track of the messages you have received.