Introductory text on Riemannian geometry

Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it is a genuine 'introduction' (as opposed to some textbooks which just seem to almost randomly put the word on the cover).

However, right from the first line: "If you've just completed an introductory course on differential geometry, you might be wondering where the geometry went", I was hooked. It introduces geodesics and curvature beautifully and is very readable.

I think the first chapter might be available on the author's website.

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.

I like do Carmo's Riemannian geometry.