I agree with many of the comments posted here that the storage strategy used by Google with Google Music is confusing, inconvenient and frustrating. The GM player itself is OK (not great), I just don't like the song management strategy.
My solution was to use the Amazon MP3 store. You can download all your purchases easily through their app and they are stored in your music folder, on your device of choice, in easy to read MP3 format. You can also upload some music for free though Amazon's limits are not as generous as Google's for space. However, anything you buy DOES NOT count against your music storage cap.
I use one Amazon MP3 account for the family (me, my wife and two sons). All of our music is there and we can download to our device or stream any of the music whenever we want. The "cool" piece about Amazon is it will allow you grant permission for up to 10 devices per account.
Another nice feature is if you use Amazon's desktop app on your PC, whenever you login to your account on your PC, it will identify your newly purchased music and download it to your PC in DRM-free MP3 format. Makes backing-up and moving music around easy.
I ramble. If you don't like Google Music, check out Amazon's MP3 store and app. I found it a much better alternative.