How to Root Galaxy SIII

Step 1. Power off your Galaxy S3.  Then hold down Volume Down, Center Home, and Power buttons together until you see the warning screen.
Step 2. Hit Volume Up button to enter Download ODIN mode and connect a microUSB cable from your Galaxy S3 to your computer.
Step 3. Next, download GalaxyS3RootNew.zip, unzip the files and run odin3v1.85.exe by double-clicking on the file. (Don’t have unzipping program? Download and install 7Zip.)
Step 4. When ODIN program opens, you should find a highlighted yellow box with a random COM number.  If you don’t see this, download and install Samsung USB drivers below:
Step 5. DO NOT TOUCH ANYTHING ELSE, hit “PDA” button and select the file CF-Root-SGS3-v6.4.tar.
Step 6. Hit “Start” button.  This will install ClockworkMod Recovery on your Galaxy S3 i9300.   If you get stuck on this step, just try a different USB port.
Step 7. Once ODIN is done flashing, you will see “PASS!” in green highlighted box.
Step 8. Your Galaxy S3 will reboot once into CWM Recovery and automatically install root and Superuser app.
Step 9. Once your phone reboots, you will find an app called “SuperSU”, this is your superuser app.
Step 10. Download and install Titanium Backup app from Play Store to verify you have root, a Superuser request window should pop up.
That’s it!  This should take you less than 5 minutes to do and does not affect your phone in any way but give you Superuser, root, and custom ClockworkMod Recovery.

I highly recommend you to boot into ClockworkMod Recovery and make your first backup ROM at this point, which will save everything including the stock ROM, your apps and settings