Every differential geometer should read at least the first two volumes of Spivak's A Comprehensive Introduction to Differential Geometry. In particular, volume 2 is an absolute gem. Not only does it reprint (translations of) original papers by Gauss and Riemann, complete with very enlightening notes and commentary, but (if I remember correctly) Spivak presents about 5 or 6 different proofs that a Riemannian manifold is flat if and only if it is locally isometric to Euclidean space. This gives the reader the best, most intuitive grasp of the concept of curvature that I have seen anywhere. (I am a firm believer in learning by repetition...)
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