How to root the Galaxy S5 on Verizon/AT&T, or just about any Android phone easily with the new Towelroot
S4 Active, Nexus 5, and on the AT&T/Verizon/Sprint Note 3. As mentioned, you are likely to shoehorn the root method on most Android handsets (save for the write-protected HTC and Motorola) without the most recent kernel version, and the process is very simple:Not only that, but the Linux kernel exploit he found, is possibly applicable to most Android devices with a kernel build before June 4th, when the vulnerability was patched. Besides the Verizon S5, the so-called "towelroot" tool has been tested and working on the AT&T version, as well as on the Verizon's
1. Download the Towelroot apk file indicated with the lambda sign on the default website;
2. If you went to towelroot.com from your mobile device, just install the downloaded APK file - remember to check the "Install from unknown sources" option from the developer menu in your phone or tablet's settings;
3. If you downloaded on your computer, sideload the Towelroot APK, and install it on your handset;
4. Run the Towelroot app, and click on the "Make it ra1n" button in the middle, then wait for about 15 seconds until the phone reboots, and you are now free to remove bloatware, ads, or TouchWiz altogether, as rooting will open the floodgates of custom ROMs and tweaks very soon.
5. If you have a kernel with a date later than June 3rd, when the latest patch was released, just install a previous kernel for your device, and you'll be able to utilize the Towelroot goods, too.
source: Geohot (XDA-Devs)