Alright, since this will be your first root, you will have to do a bit of reading before doing anything. Before beginning, you will require Odin v3.X.X (where X can be any latest version) installed on your computer. Google Odin and download it from Android Geeks (should be the second link, the first one will be of Wiki, linking to the Norse God). Secondly, you will require to have the USB drivers of Samsung Galaxy S3 installed on your computer. You can also google them and install them from Samsung's website. (Sorry, linking is not allowed in the forums). Lastly, you will require to download CF Auto Root from the internet (again Google it and follow the first link). Download the one for your exact model.
Now follow these instructions to the letter (From Chain Fire himself)
Unzip the attached Odin3-vX.X.zip file
- Download and unzip the CF-Auto-Root-....zip file
- If you end up with a recovery.img and cache.img file, you've extracted twice. You need to end up with a .tar.md5 file - don't extract that one
- (USB) Disconnect your phone from your computer
- Start Odin3-vX.X.exe
- Click the PDA button, and select CF-Auto-Root-....tar.md5
- Put your phone in download mode (turn off phone, then hold VolDown+Home+Power to boot - if it asks you to press a button to continue, press the listed button, or run adb reboot download command)
- (USB) Connect the phone to your computer
- Make sure Repartition is NOT checked in the Odin program
- Click the Start button
- Wait for Android to boot
- Done
- NOTE: Sometimes the device does *not* boot into recovery mode and root your device. Just do the entire procedure again if this happens. If it still will not install root and such, make sure that in Odin "Auto Reboot" is not checked. Then after flashing, pull the battery, and boot with VolUp+Home+Power button to boot into recovery manually. This will start the install process.
This will allow you to get root on your device quite easily. After you're done with that, you can start removing bloatware from your phone and/or download Root apps that will change the appearance of your device. As for downloading and installing custom ROMs, well that is another process entirely, which I will suggest you go after only after you have rooted your phone correctly.