Every once in a while, I get a Notification on my iPhone (SE running 12.1.2) lock screen. I see the Preview, but often the whole message doesn't show. Or even if it does show the whole message, when I hit the Home button it goes away. Well, sometimes I want to refer back to it again, but I can't find it anywhere. Do Notifications exist somewhere retrievable on an iPhone? If so, where?

It probably varies by the individual app settings (which you can change). Check in your Settings for Notifications, and go to the app you want, and make sure "Notification Center" is enabled. The Notification Center is accessed by swiping down from the top of the screen.