There are some links here:
A recent formulation of the theory is presented here, by Rovelli:
Another thing to look at, and see if you can find parts of it you can understand, is a 90-page account of recent spinfoam approach by Livine:
On the whole I think it is too hard to be an introduction for beginners, but you should know it is there (and a few parts might be helpful.)
There are simpler easier introductions than this, by other people. One for example by Hanno Sahlmann, he is a nice bright young fellow who got his PhD under the direction of Thomas Thiemann. As I recall he uses lots of pictures. I think his presentation is perhaps not so complete and not so hard as the one by Rovelli.
Do you know how to use ARXIV.ORG? To find a Sahlmann paper you go to
and you click on "search"
and you type Sahlmann into the author box.
It will give you Sahlmann's papers and there will be a recent one called something like "LQG, a short review"
There is a Sahlmann VIDEO about "new insights" in LQG. He was asked to give an opening talk at the regular Loops 2011 conference. Google "Loops 2011" and you will get the conference website, and click on "scientific program" and you will see a link to Sahlmann's talk. It might be helpful to watch the video. It reviews very recent work. He is a good clear presenter.
There is also a video talk by Rovelli at the same conference. You will see how to click on it if you decide you want to watch. It will be harder and more complete, but might still be helpful.
If you have any trouble, ask for help.
To save you time here is the link for Sahlmann "LQG, a short review" http://arxiv.org/abs/1001.4188
There are other introductions for beginners, like that, by others of the young people.
For example there is this:
called Introductory Lectures to LQG, by Dona and Speziale.
If Sahlmann is not right for you, and if Rovelli is too hard, then ask again and I or someone else will find other links. My personal opinion is that Sahlmann 1001.4188 is not up-to-date. I would only use something from 2011. But that is partly a personal bias of mine, and you might find his treatment to be just right for you. One has to begin somewhere!