For the philosophy of Mathematics side, rather than the set theory side, I'd suggest:
Philosophy of Mathematics: Selected Readings; by Benacerraf & Putnam
From Frege to Godel: A Source Book in Mathematical Logic; edited by Jean van Heijenoort
Logic, Logic, and Logic; by George Boolos
Fixing Frege; by John Burgess
Foundations without Foundationalism; by Stewart Shapiro
New Waves in the Philosophy of Mathematics; edited by Linnebo and Bueno
For their historical interest:
Foundations of Arithmetic; by Gottlob Frege
An Introduction to Mathematical Philosophy; by Bertrand Russell
One book on the set theory side that I can recommend:
Set Theory and its Philosophy; by Michael Potter