The most user-friendly and elementary introduction to Differential Geometry is Loring Tu's An Introduction to Manifolds (2nd Edition) which really tells you how to make effective calculations .
Part of the friendliness is that Tu gives at the beginning of each chapter a careful motivating explanation of its content, enriched by a short historical introduction and photograph of a main contributor.
A more comprehensive treatise which will take you further in the subject, is our friend John Lee's rich and extremely accurate Introduction to Smooth Manifolds.
It covers all the main aspects of Manifold Theory in a detailed way and has many insightful drawings, a must in a geometry book.
Do not confuse this book with Jeffrey Lee's Manifolds and Differential Geometry, which also seems excellent but which I haven't used sufficiently for me to comment on it.
The most encyclopedic treatise is Spivak's five volume (!) A Comprehensive Introduction to Differential Geometry.
I can't begin to explain why this is one othe most extraordinary books ever written in Differential Geometry and even in all of Mathematics, but let me just describe one feature.
In Volume 2 Spivak gives a 19 page translation of Riemann's inaugural lecture, which so excited Gauss, and then goes on to explain using today's notation what Riemann meant.
This is invaluable because Riemann was inventing the concept of a differentiable riemannian manifold in that talk, but didn't have the necessary tools (essentially abstract topology) to be completely rigorous.
The modern definition of a manifold was given only in 1913 by Hermann Weyl (in dimension one) and in 1936 by Hassler Whitney (in arbitrary dimension).
Similarly Spivak also explains Gauss's work in modern notation.
These "translations" from classical mathematics into modern notions and notations are just one of the gems of Spivak's masterful and incredibly original treatise.