On the music player screen, underneath the album cover on the far left of the control bar is a button to toggle shuffle on/off. If you touch the album cover the button appears for a few seconds then it disappears.
Thanks for that bit of information. You are correct in that it doesn't show up for very long. I finally was able to get it working properly by going into all settings, apps, all, google play music, and clearing the cache and data. Before that I was trying to do it under just "music" which looked the same. Thanks again.