This video hows you how to install CyanogenMod 10.1 on the *Rooted with clockworkmod* galaxy s3. Here's a link to a rooting tutorial. After that simply follow the video. The whole process from rooting to installing CyanogenMod should take about an hour. Keep in mind this will void your warranty, and I am not responsible if you brick your phone. Proceed at your own risk.
Now, for those of you who might have questions about this: What is "Rooting?
Rooting is basically getting access to the root user on the *Android* phone, similar to the administrator account on a PC. You have access to system files that a normal user won't. Simply put, root access basically makes you the "Boss" of your phone.
If you have ever owned an iPhone, you might know what "Jailbreaking" is. Rooting is very similar to jailbreaking in the sense that you can do things that you couldn't do with a non jailbroken/rooted phone. What is "ClockWorkMod?"
ClockWorkMod is a piece of software that lets you install custom ROMS, make backups of your ROM, etc. It replaces the stock Samsung recovery on the Galaxy S3. What's a ROM?
It's the software that runs on your phone. Examples are CyanogenMod, AOKP, ParanoidAndroid, etc. Will this void my warranty?
Yes, so proceed at your own risk. How do I root?
I have a link to a tutorial further down here. Why would I want to install CyanogenMod?
For one, your phone will run faster because TouchWiz isn't using up any RAM- TouchWiz isn't even there! Also, you might want a pure google experience without getting a Nexus device, so this rom does just that. Also, you might be tired of waiting for AT&T or Samsung to release their updates. With this you'll be running Android 4.2.2, so you'll be on par with those Nexus owners. What are Gapps?
They are google apps- like Play store, Google play, hangouts, youtube, etc. Without these you can't really do much with your phone. Therefore you will need to flash them after flashing CyanogenMod.
Hopefully that answers all your questions. If you have any more questions feel free to ask me!
Thanks for reading this and make sure to vote for me in the Weekend Project contest!
How to root: http://forums.androidcentral.com/t-galaxy-s-iii-rooting-roms-hacks/213815-how-root-t-galaxy-s3-sprint-forums.html
Where to get CyanogenMod: get.cm (Click on d2att and look for the 10.1.2 stable release, or if you want to use the latest nightly, that's fine too, but the nightlies are more unstable. you have been warned)
(Download the Gapps in the first row, the one labelled 10.1.x)