I'm a bit surprised not to see any Tarski references yet. Tarski's work on model theory was highly philosophically motivated and gave us the semantic definition of truth (one of the fundamental philosophical programmes in metaphysics and/or epistemology). This model theory is what put proof theory on a firm basis, gave the semantics of computation and programming languages their rigorous modern form, provided the language in which we could speak accurately about independence and other foundational relationships. It brings meaning to the possible worlds interpretations of modality, and actually gives meaning to the idea of mathematical interpretations in full generality.

If you are going to get into philosophers like Putnam and Kripke, Tarski is the prerequisite. Physical or experience-based foundationalists like the computational constructivists and the ultrafinitists are making a fundamental application of Tarski's view that model theory was actually the foundations of science in general and staking out positions that the meaning of mathematical statements must be found in experience. Tarski (as many of the Lvov-Warsaw logicians) was heavily influenced here by phenomenology.

Also, I would look at contributions from Feferman, Hintikka, Hellman, Zeilberger, and other modern foundationalists who also appear to be absent from the responses so far. I don't see how one can understand the modern philosophy of mathematics without understanding the very thoughtful approach of the various heretical controversies like Predicativism, Intuitionism and other schools of Constructivism, Ultrafinitism, etc.

Anyway, some suggestions:

Also, I think anyone who takes the philosophic foundations of mathematics seriously should invest some good time with the Lvov-Warsaw logicians, who carried out some of the deepest early 20th century analysis here even as Gödel, Zermelo, et al. were transforming the foundations. Janiszewski, Leśniewski, Łukasiewicz, Mostowski, and others explored much of the ontological crises of mathematics in ways that focused a lot of early set theory and have influenced incredibly the modern investigations.