I've been rooting and installing custom roms on Android phones since 2009. In general, it's been a fairly standard procedure. With the Moto X 2013 (and this may apply to the 2014 as well—I don't know—I can vouch only for the 2013), the procedure is slightly different.
Go to the Motorola website to request to unlock your phone's bootloader. If you got a phone not locked to a carrier, there should be no issue with this.
The first time you unlock your bootloader, it will erase everything on your phone. Back up important stuff before you unlock the bootloader!
I'm not sure why, but apparently Motorola has its own fastboot, so you're supposed to use that one instead of the regular fastboot you'd find in the Android SDK (Google has also switched things up so if you go to get the Android SDK, you'll get an Android IDE app instead of just the raw files).
The weird thing is it's very difficult to find this Moto Fastboot anywhere on the Motorola website. I've found it only via third parties.
I wasn't able to track down Linux versions, but they supposedly exist.
Next, you want to download the TWRP (Team Win Recovery Project) recovery for the Moto X 2013 (codenamed ghost). You can find the latest version on the TWRP website.
For this example, I'm going to recommend the Nexus Experience, but you can pick whatever rom you want. Unless you know you don't want Google Apps, be sure to download not just the rom but also the Google Apps .zip.
So one file should look something like Android5.1.1_NX_R9.4_MotoX2013.zip and another something like NX-GAPPS_L_Release3.3_Full.zip.
Plug your phone into your computer, and then copy those two files to the top-level /sdcard directory on your phone.
I'm sure there's a Windows equivalent for this using DOS, but I've done this only on a Mac, so I'm not 100% sure on the Windows procedure.
For Mac, put the TWRP file (something like twrp-220.127.116.11-ghost.img) in the same directory as your moto-fastboot-osx64 file (may be slightly different for Windows and Linux users, of course). For simplicity's sake, rename the TWRP file to be twrp.img.
Turn your Moto X 2013 off. Unplug it from your computer. Then, while holding the volume-down button down, press the power button. Don't let go of the volume-down button until your phone boots into fastboot mode.
Then, plug your phone back into your computer via USB.
In the Terminal.app (similar but different for Windows and Linux), navigate to the directory where your moto fastboot and twrp .img files are. If you don't know how to do that, type cd in the terminal (with a space after it), and then drag the folder over to the terminal. Then hit Enter.
Then, run the commands
You should still be at the fastboot screen, press the volume-down button once to highlight Recovery, and then press the volume-up button once to select it.
You should then see the teamwin logo.
After that, you'll get some touch-screen options.
First, select Wipe to wipe your current installation (you already backed up before unlocking the bootloader, right?). Then do a swipe to factory reset.
Once you're done, hit the back button to go back to the main menu. Next, select Install. Find your main rom (e.g., Android5.1.1_NX_R9.4_MotoX2013.zip) and flash that.
Go back to Install and flash Google Apps if you want (e.g., NX-GAPPS_L_Release3.3_Full.zip). If you don't know if you want Google Apps, flash it, just to be safe.
Then, go back to the main menu and select Reboot and then System.
It may take a while for your rom to boot up the first time, but then you're all set to use your custom rooted rom!