Everything described here is risky. You can render your phone useless by rooting it or installing a third-party operating system, and you do this entirely at your own risk.
If you’ve come to this page, you have TWRP installed and if you start your phone in Recovery Mode, you will see the screen on the left. If not, you need to read this page first. I rooted a phone! And it works!
You need to Reboot your phone into System mode. In other words, just the normal startup. If you are on the TWRP screen, just select “Reboot” followed by “System”. The next job is to transfer the other files we collected into the root folder of “sdcard”. That’s the internal memory, not “extsdcard” which I’ve assumed you removed.
There are a few ways of doing this. If you’ve ever had trouble moving files from your computer to the phone and back, you can use “Airdroid”. If not, you can connect the phone to the computer by USB and move the files using your file manager. Other ways involve using a local wireless connection, or via Google Drive, etc. If both phone and computer have DropBox installed, that’s another way. Whatever methods you have used to move stuff back and forth between your computer and your phone, will do. Just make sure you add the files to the “sdcard” top level folder on your phone. Once you’ve done this, you can switch off the phone and start TWRP (Volume-Up+Home+Power, remember?).
The TWRP Install Screen
If you hit the icon “Install” on the main screen, you get a list of folders, but scroll the screen down, and at the end of the list, there is “superuser.zip”. If you tap and install this one, your phone will be rooted. You need to do that first. You could exit out here, and have a rooted phone, but it will still be full of Samsung Bloatware. You can then install programs to enable you to uninstall unwanted stuff, but it’s risky, as your system may depend on one or two of them. So next, back into TWRP and install “cm-(version number).zip”. This will replace the Samsung Android Jelly-Bean (4.2/4.3) with the CyanogenMod Android KitKat (4.4.4) or later. If you start your phone normally and have a look at the apps now, you’ll see very few indeed. In fact you won’t even see Play Store, without which it’s difficult to add programs. So finally, back into TWRP and install “open-gapps(version).zip”, and now you will have a true Android phone, with basic google apps, and you can go to Play Store and add your own.
“Snail Through Treacle” no longer!
Now that you are rooted, it’s easy to mess up your system. I’d recommend that you take advantage of being rooted by installing AdBlock plus, this is blocked from the Play Store (for obvious reasons) so you’ll have to download and install it manually. Go to this page and read the small print under the “Get Adblock Browser” logo. Unless you know what you’re doing, leave it there. Other programs may ask permission to act under root permissions. Unless you’re confident, don’t you could kill your phone. Do your own research, on every program you install that asks.
However, what you do have, is a “new” Samsung phone that runs faster, uses less battery, and looks and feels, on the inside at least, like a brand new phone. My S3 used to run like a snail crawling through treacle, I often missed calls because I couldn’t clear an app off my screen quickly enough. Now everything I do is more or less instant! Sadly, it’s impossible to change the hardware of the phone to 4G. But in every other respect it’s great.
Oh and one other thing! As a Linux user, the updated operating system means that I can access the phone properly over USB to my computer. The original S3 hardware allowed me to see that the phone was connected, and even the folders on it, but never any files! For someone who likes to transfer files between a computer and a phone, without having to use Airdroid, or Dropbox-type cloud storage, this is a real boon!