UPDATE: This root method also has been verified to work with AT&T Galaxy S4 on MF3!
It involves an external microSD card that must be formatted in exFat format (you can do this easily on Windows by right-clicking and selecting format, you should be able to choose exFat as for format type), download and unzip the script, then copy all the files to your microSD card.
Download root script for VRUAME7 (Verizon)
Download root script for MF3 (AT&T)
Next, go ahead and download Terminal Emulator app from Play Store, open and type:
Once that’s done, simply download/install SuperSU from the Play Store, let it replace the Superuser app and also update the binaries. Reboot and you should have root!
Once you have rooted your Verizon Galaxy S4, you can follow Loki method to install CWM or TWRP recovery to install custom ROMs.
I should have an updated video tutorial but in the meanwhile, help yourself!
What is a Custom ROM?
A custom ROM is a custom, modified operating system that you can install on your Android device. Basically it's like installing Windows OS on a Macbook. By installing a custom ROM, you can bypass stock factory settings like locked hotspot tether, install new Android versions early, or even convert your phone into another OS.
With a custom ROM, you can really maximize the capabilities of your smartphone by adding custom features that have been developed by open source developers.
How to Install Custom ROMs
First, you will need to root your device, if it is not rooted yet, follow our root tutorial and root your phone.
Once rooted, you should have TWRP installed on your phone. Installation of a custom ROM is easy if you learn the basic of it. If you don't know how to install custom ROMs using TWRP recovery, see our tutorial on How to Install Custom ROM using TWRP!