There are tons of tutorials out there on how to get it in the beta stage (via invite) but since the launch today, those don't work as efficiently because the invites are gone.
Here is how I got it to work for someone else tonight...I am sharing it with everyone so hopefully it'll help some people out (sticky if someone asks?)
Some of these steps will look similar to the beta version for this...but this is the only way I could SPECIFICALLY get it to work (other proxies/methods seem to not be reliable)
Download Tor Browser (https://www.torproject.org/index.html.en)
After extraction...click "Start Tor Browser.exe"...give it a few seconds while it properly connects to a proxy and such. It should automatically connect to a US proxy (go to whatismyipaddress.com to check where it has connected. If it hasn't, you can disconnect and try again OR change the settings so that it does connect to a US proxy (a number of guides are online on how to do this)
go to google.music.com ...and sign in with your regular account. Click accept on any terms and service and acknowledge that you are going to a secure site (or something to that nature). You will get a few errors after you sign in since the browser limits a lot of executions from websites. On the main google music page, just click on the "S" with the dash through it on the lower left hand side (under "playlists"...click on "ok" for the next pop up.
Click on "UPLOAD MUSIC"...then download the music manager. Install the music manager and just for brevity's sake, select a folder with very few songs in it to upload (so that you can get started right away)
Once it's done uploading, click on "home" under My Library and voilà..your songs should be there. After this, you NEVER need the Tor browser again...any time you log in to music.google.com, it should work on your regular browser because it acknowledges you uploaded songs...but if you click on "shop" to shop for some music, it will still block you because of the browser's limits...read on to get by that
Download the google music apk (I've attached it to thise post) from these forums in order to sync with your device
NOW the fun part...getting to the Google Music store and getting some songs!
install hotspot shield. google it to find it...some people say this only half works for them as a VPN...but it's worked on every computer i've tried and is one of the few VPN's that will give you full functionality in a browser
connect to hotspot shield and use the browser of your choice (chrome, firefox, IE all work with my tests)
go to music.google.com...log into your account. Click on "shop". You will now be sent to the music area of the google market. Very fun.
This is the easiest way for the free songs...just go to any free song, click to buy it ("free" button)..and it'll ask you to log in a few times probably. It will then ask you for a credit card..put in a real credit card number (preferably yours) and whatever address you want. Just pick a US state and a valid zip code in that state (google whatever area you want). The address does not have to match the credit card for free songs. You are then given the option to complete your purchase...and then it goes through and syncs with your google music account automatically. Very sharp feature...Google is amazing.
Now the only caveat or thing you might be wondering..."what about songs that actually cost something?"...damn you for noticing the only thing i didn't cover!
The only way to bypass this is to buy, via online, a US visa or mastercard gift card that you can either reload or purchase once and throw away and then use that card to purchase the songs.
So there you go...a very nearly full functional use of google music....let me know what you think and if it's helped you out. Feel free to share the guide...but give reps my few hours of playing with proxies/tests to get it all working in one swoop. Thank you/gracias/merci
Edit: I've now attached the APK that some have requested..I know it was posted here and there...but at least this makes it a bit more of a "one stop shop" for the guide