You should be able to get a copy of both sent and recieved text messages if you go to your local store and ask an agent directly, though I must say, they seem a little reluctant to respond to queries readily without a script.
If you haven't already signed up for MyVerizon at verizonwireless.com, then set up an account for your phone number. Then, log in and go to the Overview page. If you are on a family share plan, you need to be logged in as the primary account holder to see detailed information for all lines on the account. Secondary lines will see a summary for their usage only.
For the current billing cycle, on the homepage is a link called "Current Usage" - there is a drop down for the numbers on your plan (if it is a family share). Once you click "Current Usage", there are tabs for Voice, Messaging, and Data, and within each of those is a "Details" link. There you can see all the numbers called and calls received, as well as numbers texted to and from. There is no way to see the content of the text messages, just the numbers that were involved.
If you are going back any further than the current billing cycle, you will have to pull up the prior bill, and view the call details and messaging details for each of the numbers on your account.