Excuse my ignorance if this has been addressed, or the answer is so obvious that it doesn't warrant it's own thread, but is Verizon charging a fee to use the visual voicemail? Or is that a feature of the iPhone, not something VZW has control over?
Tap Phone, then on the bottom right, you'll see Voicemail. When you have messages, they'll be listed individually for you to listen to. Much easier than the regular call-in system.
It looks like visual voicemail is free on the iPhone. I just got a letter from VZW about new features on my account, and Visual Voicemail was listed at $0.00 - I believe the free VVM on the iPhone is the reason why you lose your saved voicemails from your old phone, and they warn you about this. It is also the reason you have to set your voicemail up again.