Beginner's book for Riemannian geometry

I'm a fan of Lee's Riemannian Manifolds: An Introduction to Curvature. It is definitely an introductory book; there are many deeper topics that it doesn't mention (compare to Peterson's Riemannian Geometry). Here is an excerpt from the preface:

"I have selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject."

One of the features that I really like about this book is the careful treatment of tensors and tensor fields (chapter 2). Understanding exactly what these objects are is one of the potential obstructions to learning Riemannian geometry.

By far Gallot et al is a very good choice.

Have you tried Riemannian Geometry: A Beginners Guide, by Frank Morgan?