Songs from Google Play don't show up on phone
I found this link helpful for downloading Google Play music onto my Samsung Galaxy SIII.

https://support.google.com/googleplay/answer/1250232


Navigate to Google Play Store
Select Music
From the left soft key on your phone, select My music
From the upper left menu in Google Play (it should say Listen Now), select Playlist
Scroll to find the Free and Purchased playlist and select it
On the right side of Google Play, below the playlist image you should see a thumbtack in a circle. It will probably be grey.
Make sure you are connected to WiFi (to avoid using data) and select the thumbtack. It should turn orange and your download should start.


I have not found a way to download individual songs yet...