This is the method I used and it was awesome. Took less than 10 minutes and 5 of those minutes were used for me to contemplate yes or no to root.
[Tutorial] Rooting the Sprint GalaxyS3 the Easy Way - xda-developers
You think S3 is awesome now????...wait until you root your phone. I still find things about my phone that still fascinates me.
FYI, I use TWRP for my recovery, CyanogenMod 11 running
cm-11-20140217-NIGHTLY-d2spr.zip and it is by far the best. Everything works flawlessly, IMHO. I have not run into any issues with it yet. I also use KK Launcher Prime since I can manipulate every single thing on my phone
If you run into any snags whatsoever, there are tons of people to help on these forums or XDA forums. You will be nervous at first, but once you commit, you will be the most happy person out there. Good Luck!!! My disclaimer...
I am not responsible if you screw up your phone...
...that is why there are excellent instructions available. Very hard to screw things up if you follow instructions.