Have you set up your voicemail mailbox? The first time you dial it from the phone you should be prompted to set it up. Holding "1" on the keypad should auto-dial your VM.
If that's not it, you might need to do a simple reboot of the phone. You'd be surprised how many glitches that will sort out.
Finally, a backup and full restore might be in order. Once you do the backup a restore is easy. I like doing backups on a computer (iMac) to do the restore as a local copy of your data seems to work more quickly than going through iCloud. Both work, but local seems faster to me.
I have had my phone since the beginning of the year. Voice mail has been working just fine. I did the reboot. First time it did work. Then voice mail stopped again. i do not have a home computer connected to the internet. In a rural area and use the phone hotspot. I am not that tecky and really don't want to do a restore myself. How much luck will I have taking it in to a verizon shop?