The way into logic, Gödel and Turing I have always read about the geniuses of Alan Turing and Kurt Gödel . Many websites mention their works in logic as revolutionary. I want to understand their works, but I don't exactly know the way through which I should go in order to understand their work. To be precise, I want to know the prerequisites required to learn their theories.  I hope that members here can suggest the correct way.
 A: For a (freely available) route into Gödel, you could always try my notes, Gödel Without (too many) Tears.
A: The website www.logicmatters.net has a student guide to teaching yourself logic.  
There is also a free textbook called "A Problem Course in Mathematical Logic" by Stefan Bilaniuk if you can find it. The link I had to it seems to be broken.
A: I suggest starting with Godel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter, if you haven't read that book yet.
I think it's a good first option because there's a nice curve to the book as ideas become progressively more intricate. Even if you can't get everything towards the end or can't manage to read it all the way through, I guarantee that you will get a lot out of it.
A: For a very gentle history and introduction to Leibniz, Gödel, and Turing, I recommend Martin Davis' The Universal Computer: The Road from Leibniz to Turing. As I recall, it was accessible to a motivated high school student (U.S.) but managed to convey the basic logic, computability and mathematical ideas. It's a short, easy, fun read even if its target audience is below your level. By the way, Martin Davis is more than qualified to present this material.
