Does the music player not scan for alternative music locations?
Create a folder called music on your sd card and see if the player reads it.
I rooted my S3 a couple of weeks ago and it was easy enough. You need to be on windows to run ODIN though although I used a VM on osx.
You need a root for the specific baseband of your phone. Go to settings / About Device and you will see something like I9300XXEMB5.
Google for a root specific to your baseband. I think your handset HAS to be a factory unlocked unit as well.
Dont forget to enable USB debugging in settings/developer options
Capiche ?
Stu