Which text messaging app are you using?
If you go into your Settings >> Apps menu and open up the entry for your chose text messaging app, there should be options in the Notifications section that might need to be changed.
Check the Notifications menu in your Settings menu too, sounds like the Notifications tied to the lock screen need to be changed.
I'm using the stock Samsung messaging app.
In the messaging app, notifications for new messages is on.
In settings, notifications for lock screen are on.
I was able to get the screen to turn on when I get a new text message by using aod notify app, but it only helps if I'm looking at it because there's no notification sound and yes, I have a notification sound selected, which I hear if the messaging app is not open.