*DISCLAIMER* - EVEN THOUGH THERE IS A VERY MINUTE CHANCE OF ANYTHING HAPPENING TO YOUR DEVICE, I AM NOT RESPONSIBLE FOR ANY DAMAGES THAT MAY OCCUR. FURTHERMORE ALL OF THE PROGRAMS USED IN THIS TUTORIAL BELONG TO THEIR RESPECTIVE OWNERS AND I DO NOT NOR DO I CLAIM TO OWN ANY OF THEM.
Now that we've got the boring stuff out of the way, lets get on with the tutorial.
This exact method that i will be showing you is the method that I used to root my own Samsung Galaxy S4 on Verizon.
First, you will want to download all of the files needed. I have created a zip file right here containing all of the programs / files that we will be using.
Step 1: First things first, were going to enable 'USB Debugging' on our S4. Go into the 'Settings > About phone.' Find the build number and tap it repeatedly until you get a message saying 'You are now a developer'. Go back into the settings and you should see 'Developer options' right above 'About phone'. Click on that. Once your in developer options, scroll until you see a 'USB debugging' option. Tap on it and hit 'OK' on the warning prompt that appears.
Step 2: After that switch over to your computer. Unzip the zip file that you downloaded earlier. Run the 'SamsungDrivers.exe' program to install the necessary drivers. Extract the file titled 'Odin3_v3.10.7.zip' as well as we will be using that to flash our kernel later on. Also run the 'mobilego_setup_full818.exe' because that will be the program we will be using to obtain root.
Step 3: Now go back to your S4 and shut it down. Once its shutdown, hold the 'Volume down', 'Home button', and 'Power button' to boot your device into download mode. When you get to a confirmation prompt, click the 'Volume up' button to continue. Once you see the android guy and text saying "Downloading..." your at the right place. Plug your phone into your computer and launch 'Odin3 v3.10.7.exe' from the zip we extracted earlier. You should see a message saying 'Added!!' If you don't see that message, you probably didn't install the drivers correctly. Click the button that says 'AP' and navigate to the unziped file and find the file titled 'OC1_kernel.tar.md5'. Select that file. Once Odin loads the file, click the 'Start' button. The flashing of the kernel should only take a few seconds and you should see your device reboot like normal.
Step 4: Now that you are rebooted MAKE SURE TO unlock your device. The MobileGo application on your computer should automatically sense your device. When the little alert windows from MobileGo pops up, click the 'Manage' button. It should start a whole process during which a confirmation with a bunch of random looking characters will probably popup on your S4. Click 'OK' when that pops up. Once that happens then just wait. The MobileGo application will be flashing an app to your device. When it does there will probably be another popup on your device from google. Just click the 'Decline' button and keep on letting it do its thing. When it's done, you should see your device in the MobileGo application along with a button that says 'One-Click Root'. Click on this button. While it might seem like it's taking a while, it's just running the rooting process. After a few minutes, you should get a popup on your computer saying that root was successful.
Step 5 (optional but HIGHLY recommended): Install Supersu. For those of you that don't know, Supersu is a root manager app than makes sure you are asked before an app gets root permission. To install it it's as simple as going to the play store and downloading it. Once you've downloaded it, open the app and it will ask you what recovery you have installed. Click the 'Normal' button and let it run. It should say that it was successful and from there hit the "Reboot" button to fully apply it.
Step 6 (optional but also HIGHLY recommended): Flashing the newer kernel. By flashing the newer 'OF1_kernel.tar.md5' you will not lose root and you will get rid of a very pesky message that pops up every time you do something with root on the OC1 kernel. There really is no downside to it and it only takes a minute. Repeat 'Step 3' but instead of selecting the 'OC1_kernel.tar.md5' choose the 'OF1_kernel.tar.md5'. Once it flashes your phone should automatically reboot and be rooted on the latest kernel.
All comments, feedback and questions are appreciated as this is my first post ever to XDA Forums.