Assuming you would like a step by step process on how to sync your google play music onto multiple devices:
The first thing you need to do is download the Google Play Music app on all the devices that you want synced. Once the app is downloaded, open it and sign in with your Google account. Once you are signed in, tap on the menu icon in the top left corner of the screen and select Settings. In the Settings menu, scroll down until you see an option called Download Location and tap on it. This will allow you to change where your downloaded songs are stored on your device. You want to make sure that this setting is the same on all devices so that they can access the same files.
Next, go back to the main menu and select My Library. Here, you will see all of the music that is available to you through Google Play Music. You can either start playing a song from here or add it to a playlist. To add a song to a playlist, tap on the three dots next to it and select Add to Playlist. If you want to create a new playlist, tap on the + icon in the top right corner of the screen and select Create Playlist. Give your playlist a name and then add any songs that you want included in it.
Once you have created your playlist (or if you just wanted to play a song from My Library), find it in My Library and tap on it. This will take you into playback mode for that particular playlist or song. In playback mode, there is a small down arrow next one of the playback controls. Tap on this arrow and select Available offline. This will cause all of the songs in that particular playlist or album be downloaded onto your device so that you can listen even when there is no internet connection available.
Now, whenever you make any changes to your library (adding new songs, deleting old ones, etc.), those changes will automatically be reflected across all of your devices since they are all synced together!