To save voicemail messages: Welcome Verizon Wireless Customers: CBW Productions
That's assuming you have the basic voicemail, and not visual voicemail. With visual voicemail, you can just save the sound file.
As far as pics, that depends on the type of phone. A basic phone, either text them to your email address, or backup to an SD card if the phone is capable. On a smartphone, backup to either Google, iTunes, or OneDrive, depending on the OS of your phone. Or an SD card if the phone accepts one. (I left out BB, and their proprietary backup system)
You can also hook the phone up to your computer via USB and copy the files from the phone to your computer. Exact steps will vary, depending on the make and model of your old phone.