Apple - How can I hide the onscreen keyboard in the iOS messaging app?

This is not possible at all right now. Like you said, you can back out and go in again to dismiss keyboard. Other option is to tap the 'edit' button on top right. That will dismiss keyboard but it will go in edit mode. When you cancel out of edit mode keyboard will come back.

Apple should do something to easily bring up and dismiss keyboard...

Update: With iOS 5, you can swipe messages down all the way pushing the keyboard downwards. Initially I thought, "why not just swipe down the keyboard itself to dismiss?" But now it makes sense because you start scrolling down to see previous messages and keyboard naturally goes away when you do that.

If you turn your phone sideways, the keyboard is slightly different and has a few different command keys on the sides. The one next to "return" will hide the keyboard. Then simply turn your phone back up and the keyboard stays hidden until you click for it to come back. I found scrolling up in the message will hide the keyboard. But this only works in the SMS screen, it does not work in apps like Google Docs, whereas turning it sideways does.

Found new trick in ios 13 to hide the keyboard.

1. From the page where you want to hide the keyboard, open notification area by sliding the screen from top.
2. Swipe right to go to Search panel from notification area.
3. Click on search input box.
4. Now just swipe up the screen from down to close search panel.
5. now keyboard will be disappear from the application.