[T-Mobile, Samsung Galaxy SIII] blu422:
Looking into rooting? Maybe try this foreign ROM thing? (Not talking about ethnicities here) Here is the place to be.

First of all, make sure you have a T-Mobile Galaxy S3. (Aka t999 or d2tmo)

If you follow instructions here for any other device, even for galaxy s3's of another carrier you are extremely likely to regret it and end up having to unbrick your device because you installed an incompatible file.

You can take your battery out to make sure it says t999 under if you are not absolutely sure.

Rooting is a similar idea to jail breaking an iPhone. It removes what restrictions there are with android devices. Yes, it is perfectly legal.

Which leads me to my disclaimer:

I take no credit for what's posted here.

The only work here that's mine is the organization. Please donate to the rightful people who spent quality time developing.

Also I am not responsible for any damage done to your device.

It is vital that you follow instructions in every exact detail.

Firstly, rooting.

1. Install drivers for your device onto your PC. To do this, you can install Samsung Kies or there are downloads that will install the drivers solely(recommended link for drivers http://d-h.st/zya)(thanks DocHoliday77). If you get to the point where you can exchange files to phone from the computer and vice-versa then you're set for drivers.

2. Do a backup (using Kies or a program of your choice)

3.Uninstall Kies. Kies doesn't play nicely with the program used to root called Odin. So uninstall Kies so Odin can work unhindered. Your drivers should stay. But make sure you can still copy files between your phone and computer before proceeding.

4.Follow these instructions put together by OfficialPhoneRebel(aka EverythingApplePro):

http://www.youtube.com/watch?v=fKwhAVj5Feg Outdated. try CF-Auto-Root

Yes, I used this method and it did work for me.

5.After you are rooted, the first thing you wanna do is either install ROM manager (on play store). Open ROM manager. (Yes, give it superuser rights) and install the latest clockworkmod recovery, or use GooManager to get TWRP recovery, both work, but sometimes one is suggested for the installation of a specific ROM.

6.Now install busybox(also in play store)

7.You're all done!

If you've made it this far, kudos. Now you're gonna wanna get the app called titanium backup and do a full backup of your phone. Also boot into recovery (home+volume up then press power until it vibrates all while holding home+volume up buttons) do a backup through these menus. If your feeling like an over achiever and wanna be super safe then go ahead and back up to both your internal sdcard and your external sdcard. Also it now would be wise to back up your IMEI nv (basically your phone's identity), instructions for that can be found here (thanks to daxecutioner24)

Now you have reached a possible stopping point. Stop here and enjoy root privileges. Continue to read how to install a custom ROM.

Installing a custom ROM(operating system):

GooManager (on play store) is a pretty good directory for ROMs. And is a great place to download them and gapps. Now since not every ROM installs this way it is important to check the main page associated with that page to see if any special instructions are required.

But this is what you generally do.

http://xdaforums.com/showthread.php?t=1946701

Roms:

Gapps can be found either on the ROM's page or through goo manager make sure the firmware versions match up (like android 4.1 and 4.1 jelly bean gapps)

For a pure android ROM I would suggest CyanogenMod 10.1 (the one I am currently on)

For an inverted stock ROM stuffed with fun features I would suggest DaRTHSTALKER S3

For a full list of roms and descriptions or if your just addicted to flashing, then here's the stockpile of almost every rom made for the galaxy s3 t999

http://xdaforums.com/showthread.php?t=2088179

Backups are your friend. Especially through recovery mode. They have saved my butt several times when flashing a ROM went wrong.

And yes it is possible to sim unlock (make it so you can use it on any GSM network(such as at&t or US cellular)) your phone:

http://xdaforums.com/showthread.php?t=2037390

Problem happened and you can no longer boot correctly? This is actually very common.

If you can boot into recovery:

Boot into recovery (Hold home button and volume up then press power till it vibrates all while holding volume up and home)

Restore backup (easiest method)

If that falls through then install a rom by following the instructions mentioned earlier.

If you can't boot into recovery or if those methods didn't work:

You must odin to stock. Follow the instructions here under using odin. Odin and Driver files are at the top of the post. Firmware files (what you put in PDA) are located here

If this didn't work, you may be out of luck, try again and if you are still stuck browse the forums and possible post something under Q and A if worst comes to worse, make sure to describe the entire situation including error messages you may have received so the problem can be pinpointed.

And finally, everyone be nice seeing as I am welcoming very new people to this thread.

Also if any suggestions to improve this post let me know.