I want to know is there a software or system that will allow me to
transfer voicemails from my verizon wireless cell phone and save them
via some other media?

There are plenty of ways:

Does your phone have an audio out jack...if so, you can patch your
phone to the pc and record your voicemails by replaying them.

You could sign up for GotVoice.com service...

You could use SKYPE (Skype.com) to call your cellphone and retrieve
your messages while recording with Audacity or the like...

You could use GizmoProject (http://www.gizmoproject.com/) instead of
SKYPE as it has built in recording (you don't have to use Audacity to
record)

In fact, if you look around your PC, you should find a phone socket.
Plug your phone cable in, choose a suitable audio recorder (e.g. the
free program Audacity) and then select "Record from line in" or select
'Line in' on the recording menu.