Conics

I heartily endorse reading the classic works of geometry as a way to both the subject as well as a way of thinking about math, proof, and argumentation. Start with Euclid's _Elements_, and then move onto Archimedes' short books on levers and floating bodies, Apollonius's wonderful treatise on conic sections, and Ptolemy's _Almagest_. They are excellent for self-education, providing both geometrical knowledge in itself, as well as extensive training in sound reasoning. Don't be fooled by the antiquity of their origin: they teach more clearly than most modern day texts, and their content is timeless.