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.