As usual, while trying to use a guide to install something, I failed miserably. In this case, it was the new Cyanogenmod 9 Nightly builds for the Samsung Galaxy S3. As a result, here's a quick guide to how I got it working properly, including all the files you'll need in a handy zip, numbered with the step which requires them. Also included are some quick troubleshooting tips for issues I faced along the way.

BEFORE YOU DO ANYTHING, BACK UP EVERYTHING!
(Thanks to theandroidsoul.com for the initial guide that this one is based on.)
1) Install either the 64bit (a) or the 32bit (b) drivers to your pc.
2) Copy the Google apps and Cyanogenmod packages to the SD card of your phone.
3) Run Odin, reboot the phone while it is connected to PC and hold Volume Down + Home + Power.
4) Tick the box next to "PDA", and click the "PDA" button, select the file "4-CF-Root-SGS3-v6.3.rar" then press start. Your phone should reboot.
5) Hold down Volume Up + Home + Power until the Samsung logo appears, then release. Select "Backup and restore" and then "Backup."
6) Select "install zip from sdcard" and then "choose zip from sdcard", select the Cyanogenmod file. Once this is complete do the same for the google apps file.
Troubleshooting:
Phone isn't detected by the computer/Odin:
 
- Reinstall the drivers
- Try alternate drivers (32 bit instead of 64)
- Install drivers using Samsung Kies
- Switch USB ports
- Enabled USB debugging in phone options
- Refresh the hardware in device manager (Device Manager> Scan for Hardware Changes)
- Use another computer (annoyingly I had to do this)
Phone boots into Android Recovery 3e not ClockWorkMod Recovery:
- Re-run Odin.
- Find the most recent version of CF-Root (This was the issue for me, needed v6.3)
Cyanogenmod hangs while booting:
 
- "Wipe data/factory reset" in CWM Recovery (BACKUP FIRST!)
- Try a different Nightly
Hopefully this will help someone learn from my mistakes and save themselves a few hours.
Downloads:
Dropbox (might die if it gets hammered)