Look on the phone.
There is no way to reliably view all message content on a phone other than by looking on the phone. There are too many ways to send messages which would bypass any attempt to do so on one specific platform.
Messaging platforms would include SMS/MMS, iMessages, Snapchat, Twitter, Facebook Messenger, etc... There is not all encompassing method to view all messages from all platforms.
You can install Verizon Messages on your child's phone and IF they use it, you would be able to view SMS/MMS messages on a computer IF you have the login userid and password for their MyVerizon account. Of course, if they choose to make another texting application the default texting app, you would no longer be able to do so. If they chose to change their password, you would no longer be able to do so.