Having trouble rooting Samsung Galaxy s3

Hey. I have an unlocked Samsung Galaxy S3, by t-mobile. It was originally only a 4.1, but I had it upgraded to a 4.3, which I admit I am still unsure means it's now a jellybean, or a kitkat. I am fairly sure that I would rather it be rooted, but I tried quite a few different programs that were supposed to help me accomplish this, and nothing worked. And I have spent hours reading about, and writing info on rooting, so if I finally get it done, I will be pretty sure I know what I am doing. Anyway, the model is a SGH-T999,and I could really use some knowledgeable advice on how to proceed, so there's less of a chance of my phone being bricked, or my screwing it up in some other ways. Thank you, Vicky c.

I'll read it as soon as I get the chance to. Thanks.

Thanks, but I actually did try the Kingroot app, and I remember something about framaroot, but am unsure if I gave that a shot also. Kingroot wouldn't work, and I got frustrated and gave up after hours of trying all types of things.. That's what led me to write here about it. I looked at the site that that guy Iago suggested, and it's very informative, just seems a bit complicated. But, thanks again for answering my question.

I just downloaded those apps I spoke of on Google Play, and tried them out. I haven't rooted my phone yet, but when I do, I am most likely going to follow the directions given by some one from the XDA developer's Forum, because they seem to have a step by step process for many different types of phones, plus these people really seem to know what they're talking about. Hope that helps.