Polya's How to Solve It changed the way I thought about learning mathematics. His treatment of random walks in one dimension (eventually all walks return to the same point) vs three dimensions (where they can escape) really affected my mental model of the world.
Instead of How To Solve It, which is organized dictionary–style with short sections on particular named problem solving topics, and is somewhat hard to interpret for novices without guidance, let me recommend Polya’s other two books (each 2 volumes), Mathematical Discovery and Mathematics and Plausible Reasoning.
