To start from absolute zero, check out Suzanna Epp's Discrete Math[0]. I believe even a motivated high school student could get started with it and even finish it. If your proof-writing is shaky, the book provides a very good workout. From there it will be easy to choose the areas of discrete math to specialize.

[0] http://www.amazon.com/gp/product/0495391328?keywords=susanna...

Thanks, I'll check it out. My proof-writing is definitely shaky. I can clearly see the relationship between programming on writing proofs, but I can't get immediate feedback on the validity of my mathematical proofs like I can with code.

Questions in Epp are by no means unique. If you search MSE, you'll see that every question in Epp has probably been asked and reasked about a thousand times each. That goes for subjects like Real Analysis, Abstract Algebra, Topology as well.