Thanks to Critical Composer for this information. You can find Critical's original post here: https://forums.geforce.com/default/topic/732693/shield-portable/rooting-your-shield-the-why-and-how/
Rooting: What Is It and What Can It Do For You
Rooting the Shield brings with it both great power and great danger. You will have access to the root of the Android operating system. For those that don't quite understand what root actually means I'll break it down in to Windows terms. The internal storage would be akin to a mix of My Documents and Program Files. Most downloaded apps will store data and settings here but it is also where any downloaded files are store, whether that is mp3s, pictures, or documents. In Linux(and Android), root is one folder up from that, or direct access to the C:/ drive where you can access the users folders(including app data) and, more importantly, the Windows folder. As far as pathing goes, the internal storage is usually found at "/sdcard0". One folder up would bring us to "/" which has folders such as "data", "mnt", "sbin", and "system". These folders combined, with a few smaller ones, are what would be found in the Windows folder. Having access to these files means that we can directly alter how the Android OS operates, including changing permissions, altering what the hardware keys do, how the interface looks, or completely removing preinstalled apps instead of simply disabling them. Being able to alter this data also means that changing the wrong variable or deleted some lines of code could destroy the OS causing the user to have to reinstall. With that said, most uses for Root involve using apps from the Play Store to alter these files for us. As long as the app is reputable there shouldn't be any negative side effects to using these apps.
One piece of disclaimer though, nVidia will provide support to rooted Shields on a case-by-case basis. As a rule of thumb, hardware damage will fall outside of this support. Physical damage caused by extreme heat from overclocking most likely will not be covered by warranty support. I can't promise that all software issue will be supported but a simply reinstall of the OS should fix any of those. Again, don't mess with system files or settings unless you know what you are doing and don't install shady root apps and you should be fine. If you are unsure about something root related you can always create a topic on this board or message me and get help before messing with something.
I'm Fine With The Risk, So How Do I Do it?
Rooting is actually a 2 part process: First we must unlock the bootloader and then we must grant root access to device user. The bootloader is the software that initially loads when you turn your Android device on. The bootloader is a simple program that loads the complex program, the actually OS. If sure many of you who are still on the fence about rooting have heard horror stories of getting a device rooted. Most of these stories are based on how difficult a vendor has made it to unlock the bootloader. HTC requires the user to go through their developer channels and Samsung usually requires user to jump through many hoops to do this. nVidia has decided to make unlocking the bootloader a simple, single command line. But first we must make sure the proper drivers are installed. The regular drivers that Windows most likely installed when you first connected it do not need to be uninstalled. The drivers that need installing are the fastboot/adb drivers for when the Shield is in the bootloader menu. First lets download the files needed. You can find everything in this dropbox. This includes the Fastboot Drivers, the root images, and the files required for using fastboot. I am not the creator of these files, I simply packaged everything into an single zip for ease of use. The drivers can be found orignally at XDA and the root files from Gnurou on Github
Shield Root Files
1. Turn your Shield off completely. When it is powered down, hold Back and Home and hit the Power button. Continue to hold the Back and Home keys until the bootloader appears. Your screen should now look similar to this. Note: It won't say "Device - unlocked" and won't have the red warranty void text.
2.Plug your Shield into your PC via the USB cable. You should receive an error that says the drivers could not be installed for this device. Open Device Manager(hit the start button and search for 'device', it should be near the top).
Under the "Other Devices" area should be a device called "fastboot". Right Click that and hit "Update Driver Software".
From here click "Browse my computer for driver software".
On the next screen you will want to click "Let me pick from a list of device drivers on my computer".
A box will appear asking what type of device, simply select "Show All Devices" and click next.
A blank box should appear on the next screen. Click "Have disk" and navigate to "android_winusb.inf" found in the Universal Naked Drivers you downloaded earlier.
Open that an a list of devices should appear in the box that was just empty. Scroll to the bottom and select "Nvidia Shield Fastboot", then click next.
A warning will appear letting you know the drivers aren't officially signed. Just hit yes to agree to continue installing them regardless.
Your drivers are now installed! The most difficult part is over now.
3. Open the Root folder now. If you hold left shift and right click inside of the explorer window, one of the options should be "open command window here". This will open the command prompt but instead of starting in C:/ it will start in the current folder. If you type "fastboot devices" it should show a bunch of letters and numbers followed by fastboot. This means that the Shield is properly connected to your computer. Before we type the command to unlock the Shield's bootloader I must place this disclaimer:
This is the part where the entire internal storage will be deleted. Make sure to back up any data that you wish to keep. I've heard Helium is a really good app for backing up data that doesn't require root. As a precaution, I always remove the SD card simply because I am paranoid it will delete that by some random act of entropy.
In the command prompt type "fastboot oem unlock". It will present some information and ask if you wish to proceed.
Press either the home or back button to navigate to 'Unlock" and the press the power button to complete the process. The screen will return to the normal bootloader screen but, at the top, it should now say "Device - unlocked".
4. Some people go straight into rooting the device at this point. I always suggest hitting continue and letting the device boot first. This will do first boot initializing and seems to cause less issues than doing unlocking and rooting in the same swoop. After your device has booted, go through the introduction menus, and shut your device off. Go back to the bootloader(Back + Home + Power). Back on the PC go back to the command prompt. Type in "fastboot boot root_shield.img". This will allow the user root access while also installing the SuperUser app to the system. After you type this in, the Shield should display some penguins across the top with some text in the middle. The device will then reboot. You Shield is now rooted.
Now you can use all of those awesome apps that require root(including fixing the SD permissions in the latest update). For a great list of some of the apps you can now use check out HankChill's posthttps://forums.geforce.com/default/topic/734204/general-discussion/rooting-your-shield-awesome-things-you-can-do/ For future reference, any time the Shield receives an update it will need to be rerooted. The update will download and install just the same as a Shield that isn't rooted, but the update will remove the root access during the installation. To reroot the Shield you only need to complete Step 4. One line of code and your Shield is rooted again. If I missed anything or there are any errors let me know and I will correct them.
(Conclusion) Clarified how updates are installed with a rooted device
(Installation) Detailed where to files originally came from and their links
(Introduction) Added information about nVidia's warranty support for rooted Shields
(Conclusion) Added link to HankChill's post about Root Apps