[AT&T Galaxy Note 3 Q&A, Help & Troubleshooting] GregJ7:
I just did this over the course of hours. I can't say I'm pleased with any of the explanations on xda-developers or found by Google. Unfortunately, writing up a good, full explanation is beyond me right now.

I did this from the Android 5.0.1? Lollipop "OC3" version (seen as the last 3 characters of the "Baseband version" under Settings -> General -> About device). Apparently this version cannot be rooted, however, you can downgrade your version of Android to one that you can root: Android 4.4.2 KitKat ("N900AUCUCNC2"). Apparently you can do something to stay rooted while manually updating to Android 5.0 Lollipop "OC1," but updating to the current AT&T version "OC3" revokes your root access.

I followed the process listed at https://xdaforums.com/note-3-att/general/root-att-note-3-android-5-0-t3318130/post65467618, but if you are satisfied with having rooted Kitkat 4.4.2, you don't need to do steps 11 and 12. If you want the more recent Android 5.0 OC1, then do steps 11 & 12.

One thing I can do is help you understand some of the things I never saw explained. In the list of instructions you will see some programs you will need. A brief summary of each:

Odin - A program that runs on Windows that will manage some low-level aspects of your phone over a connected USB cable. It is used to install an OS (Kitkat 4.4.2 in this example) after your phone has been Wiped from the boot loader menu (gotten to by turning your phone on holding down VolumeUp + Home buttons). It is nice to have a MicroSD card onto which you can move various things before wiping the phone. (You never need to wipe your MicroSD card. Make sure you don't wipe it when using the Wipe feature in SafeStrap).

Towelroot - This is actual program that roots your phone using a known problem in the phone. After running the program, tap the only button there is to do it. I think this is a safe operation—as in, I think it will tell you if it can't rather than scrozzle your OS.

SuperSU - A program that manages which apps on your phone have root access. After it is installed, when you run a newly installed program that requires root access, SuperSU will pop up a dialog asking you if you want to grant it root access. I saw no indication you can get by without it, even if you don't need to "manage" app root access.

SafeStrap - This is a program that does various things, most importantly installing other operating systems that you previously loaded onto your phone as a file and then letting you choose which one you want to boot to whenever your phone boots up. It installs like a normal program, however from within the program you can "Install Recovery" or "Uninstall Recovery." "Recovery" is a bit of software it inserts into the phone boot process. While booting, if you want to use that software, you tap "Recovery" in the lower left. Tap "Continue" if you just want to boot with the currently active OS (chosen from within the Recovery software under the "Boot" options). With Recovery, you create a new slot for an OS, MAKE IT THE ACTIVE OS, Wipe it (Advanced, NOT format), then can back up a menu and install an OS from your MicroSD card. The purpose of all these shenanigans is so you don't have to lose your working Android install while installing something else.

Titanium Backup - A software backup program that requires root access to work. It also has a ton of other features loosely related to backing up. It is the simplest way to prevent AT&T from updating your version of Android to the unrooted current version. Having said that, it is anything but simple because of the user interface. You might investigate alternative methods for preventing auto-updating.

Be aware that whatever software (e.g., file manager) you use to hunt down files and folders on your phone is probably not going to be showing you everything. There are hundreds of programs that your phone uses, some of which are hidden. Titanium Backup does appear to be capable of showing a true full list of apps (including system apps). This is pertinent, because the AT&T Software Update program is hidden very well.

Potential costs:

$10-20 US approximately for a MicroSD card. There's fast ones that are in that price range. Google for reviews.

$5.99 for Titanium Backup from the Google Play store.