Sadly - unless you have an Android powered smart phone, you can't listen to your Google music tunes while you are offline. You must be online and able to stream tunes. While getting ready for a flight to London I started looking for a way to listen to my google music tunes while i had no internet connection and found:
Melodies Pro - A Google Music Client
This $0.99 app for the iPhone allows you to quickly sync a playlist to your iPhone from your google music account. The tunes are then downloaded (with cover art and music details) right to your phone so you can listen off line. This transformed Google Music from streaming to something much better and more useful.
In practice I found it to be easy to use, and stable. On a flight back from London I was able to listen to tunes all the way home. Background playing works with no worries. I was able to modify/update the playlist from my phone via the web version of Google music - then the app simply updates and downloads the new songs added to the playlist.