Developer "Open1your1eyes0" has been the go to guy for Root on the Stock builds for the Verizon Galaxy S3. He has already achieved root on the Verizon variant of the Galaxy S4 even though we still don't have a public method for unlocking the bootloader. The guide at the source link is very simple as long as you are good with following directions.
You will need to first prepare your PC and phone for the process by installing the Samsung USB drivers and downloading a few files. You will then flash a special Kernel that removes the "Root Block". Lastly you will Root your device. There is a disclaimer attached to the guide.
If you can wait a few more days @DjBliss will be releasing his method for unlocking the bootloader. Once that is accomplished you should be able to easily flash a custom recovery, and then install superuser.zip. That would be a cleaner way with minimized risk.
Guide here via XDA
You will need to first prepare your PC and phone for the process by installing the Samsung USB drivers and downloading a few files. You will then flash a special Kernel that removes the "Root Block". Lastly you will Root your device. There is a disclaimer attached to the guide.
Warning, this method will leave you with a miss matched kernel/modules. This may cause problems unless corrected.
If you can wait a few more days @DjBliss will be releasing his method for unlocking the bootloader. Once that is accomplished you should be able to easily flash a custom recovery, and then install superuser.zip. That would be a cleaner way with minimized risk.
Guide here via XDA