I hold down power up + home + power and it brings me to a screen that says Android System Recovery - I can't get the Clockwork Recovery. What could be the problem?
You will need to flash an other package after you install CWM. The Galaxy SIII has checks in its start up scripts that if the recovery has been modified it will replace it with the stock recovery.
You can try this Root Package and flash it after you flash CWM. Besides installing SU and superuser, this removes the scripts that does the Recovery Check.
The actual scripts that are responsible are :
The package, that I linked, will install root, busybox, and it renames these files so they don't execute.
When the device boots, these scripts run and check the recovery, if you modify it, it replaces it with stock. The only way around it is to stop these scripts from running.