One more vote for Gallot, Hulin, Lafontaine. I think this book does a better job of most of presenting clean proofs (including avoiding the use of co-ordinates and Christoffel symbols) and a more geometric approach than other books, which tend to get bogged down in the abstract formal computations. A lot of important explicit examples are worked out in detail. It also shows very nicely how curvature bounds can be used with Sturm-Liouville theory applied to Jacobi fields along a geodesic to establish global geometric properties of a Riemannian manifold. This is the heart of global Riemannian geometry as developed by Berger, Toponogov, and others and raised to a high art by Gromov and Perelman among others. But you wouldn't know that from many other books on Riemannian geometry.