Download both files from the downloads section just above. Power Off your Galaxy S3. Wait for seconds after display is off. Now, press these 3 keys together until you see warning screen: Press Volume Up to continue to Download Mode. Connect Galaxy S3 to PC. Some points to be noted: If you did already, try installing drivers again, using different method as discussed above in drivers section.
Plus, after installing drivers, try these tricks: Leave everything else untouched. Odin should look like this: Your device will reboot automatically upon successful flashing, and will be rooted when it has restarted. Upon successful installation, You Odin will look like this: For this, boot your Galaxy S3 into Recovery Mode: Wait till capacitive lights of Menu and Back keys are off.
Then you can download and do it yourself with better idea of what you are going to do and how. Installation will be over soon. The program will run itself now. You can run it manually from the the shortcut at desktop it just created or directly from its installation folder. Select NO when it asks for update, and you will get a screen where you get to choose your Galaxy S3 device as per build no.
Yours is model no. Now, select the appropriate option based on build no. The toolkit might need to download the right files, as regards root, recovery, etc. Unplug the Galaxy S3 if it is connected to PC and then Select Option 1, then option 1 again on next screen and then follow the direction to complete the drivers installation.
After drivers installation is finished, you will be back on Main Menu. Now, select option 2 for Root. Select any one of the six options to root the Galaxy S3 that toolkit provides you.
Each one is explained there so that you can decide and choose. As I think recovery is a must-have, I prefer having recovery all the time and would suggest you to simply select the option 3 ALLINONE, wherein everything required will be flashed and you will have both recovery and root installed on your device.
Disconnect phone from PC. Confirm your selection by pressing OK. Connect phone to PC using the cable now. The toolkit will proceed further once it finds USB Debugging is enabled. Actually, and unfortunately, you gotta do stuff manually now. Toolkit will load Odin for you. Now, read what and what toolkit asked you to do with Odin and then select the file as it specified in odin: Click on PDA tab on Odin and select the file mentioned in point no. You must be able to locate that file in: Hit Start button on Odin now to start the flashing process.
The recovery will be flashed on your device, which will be then used to install Root on your device, which is all automated. Once toolkit has done its job, your phone will be rooted already and will have the recovery you chose installed.
Toolkit will return to Main Menu, and you close it now. Using Terminal Emulator android app to flash.