The voice mail is actually saved on the Verizon servers, not on your phone, so other than playing it back and recording the message, I can't think of a way for you to do this, at least not using standard voice mail.
On the other hand, if you sign up for Google Voice you can "activate" Google to your cell phone from the GV site, it will give you a string of humbers to enter into your dialer, and it will cause all voice mail to your cell phone to be "re-routed" to Google voice. From Google Voice you can then listen online, on your phone, even email the voice message to someone. It will even "attempt" to transcribe the voice message and email you a *rough* transcrip of the voice mail. If you install the Google Voice app on the phone, you can even play back the Google voice messages directly on the phone. (You can also call your Google Voice number at any time to get your messages.)
The service is free, but you have to request an invite at http://google.com/voice It will probably take about a week to 10 days to get one.
If you take a look on their web site and want/need an invite sooner, let me know, I have an extra invite i can send to you.