From the Google Play Music App, open up your Playlist section. Find the playlist you want to shuffle, and tap on the three vertical dots on the lower right. Choose Shuffle on the popup.
You can turn on Shuffle mode by first starting song from your playlist, and while it's playing, tap on the banner at the bottom that displaying your currently playing song. This will bring up Google Play Music's player interface, which has the song name scrolling at the top, album art in the middle, and the play/pause buttons at the bottom. Next to the play/pause button, in the lower right,is an icon that looks like two crossed arrows. That's the shuffle button. Tap to turn it orange and it's on. It will stay on even if you switch songs/albums/playlists.