I would differenciate between some books. If you simply want to learn how to calculate things, for instance because you are interested in physics, I'd recommend to you the book "Geometry, Topology and Physics" by Nakahara which has a very good part on Riemannina goemetry. However, it may contain more than you actually want to know.
The book with which I have really understood Riemannian Geometry was Jürgen Jost's "Riemmanian Geometry and Geometric Analysis". However, it might require some prerequisites that you haven't got so far.
Alternatively, you could read Do Carmo's book. It is very concisely written and features some nice examples.
I would not advise you to read the Gallot-Hutin because some of the proofs are simply only outlined. For a person that begins learning Riemannian geometry this could be very discouraging. However, I admit that the great advantage of this book is the number of exercises.

Something I want to add because I really want to advertize one book: Riemmanian geometry by Takahashi sakai. If you once have got the basics, this books takes you further. A quantum leap further. But you really need the basics. I would use this book for a second course in Riemmanian Geometry, assuming the student's familiarity with differentiable manfiolds and fiber bundles and a first course in Riemannina Geometry, such as for instance material covered in Jost's book in the chapters 1-4.

A Panoramic View of Riemannian Geometryas a textbook, but while you work through some textbook, it is nice overview of some of topics. $\endgroup$ – Ben McKay Mar 27 '16 at 10:36