Now head over to XDA-Developers to get the needed files to root your Verizon's Samsung Galaxy S3 that runs on Jelly Bean. Just give credits to open1your1eyes0 and the rest of the gang involved in making this possible rooting technique - http://forum.xda-developers.com/showthread.php?t=2046439
Here's a visual aid for beginners that might confused from the advance users guide.
Flashing the Bootchain with OdinThe first bootchain is VRALEC, you need this so that you can install the Custom recovery of your choice like CWM (ClockworkMod Touch Recovery 18.104.22.168) or TWRP (TWRP Recovery 22.214.171.124).
Assuming the you already installed the USB driver. Now run Odin then click on PDA button then browse to the downloaded VRALEC.bootchain.tar.
Boot the Galaxy S3 into Odin Download Mode
Hit Odin's "START" button to flash the VRALEC bootchain. Just wait until it finishes.
Flashing the Custom Recoveries CWM or TWRP with OdinYou need to choose the custom recovery of your choice, either CWM or TWRP. You will need this so that you can the install the SuperSU apps via SD card update zip option.
Download the TWRP or CWM onto your computer.. Now load the CWM or TWRP tar files onto Odin's PDA button. This time Uncheck the "Auto Reboot" checkbox option.
Flashing with CWM - load the CWM6012TouchVZW.tar.md5
Flashing with TWRP - load the TWRP2310VZW.tar.md5
Now boot the phone into download mode again, then hit "Start" to flash the custom recovery. Remove the phone from PC once finishes.
As soon as flashing recovery is done in ODIN and you see the screen go black, immediately press and hold Home+PowerUp before the device powers on again (have to do it quickly) and hold it until you see it boot into the CWM/TWRP screen.
Install ROOT or Unlock Bootloader
Once you already installed custom recovery is in place, put the root.zip on your SD card . . . boot into recovery mode and select the "install zip from sd" or whatever the wording is that is there . . . use the "select zip" menu and point the recovery program to the correct zip file you are installing
Download one of the below
For Root Only - Download SuperSU to your SD Card - Link
For Unlocked Bootloader Only - Download Verizon Galaxy S III Unlocked Bootloader to your SD Card - Link
For Both Root and Unlocked Bootloader ***RECOMMENDED*** - Download SuperSU + Unlocked Bootloader to your SD Card - Link
Restore the Original Jelly bean VRBLK3 Bootchain
This step is optional but recommended to return your original partitions after you see everything is working properly
Boot phone into ODIN mode
Flash the VRBLK3 Bootchain using ODIN v3.07
After a final reboot, you are all set to use your device as needed