Differential Geometry is one of the toughest subjects to break into for several reasons. There is a huge jump in the level of abstraction from basic analysis and algebra courses, and the notation is formidable to say the least. An ill-prepared student can begin reading Spivak Volume I or Warner's book and get very little out of it. This is, in fact, what happened to me. Only at the advice of a professor did I take an undergraduate diff. geometry course which used this book, and am I glad that I did.
In short, here is a book which takes the key aspects of classical and modern differential geometry, and teaches them in the concrete setting of R^3. This has several advantages:
(1) The student isn't lost in the abstraction immediately. When I took my first diff. geometry course, we spent the entire time taking derivatives in n-dimensional projective space and other equally abstract spaces. This book keeps it concrete, and supplements each idea with several worked out examples to help ground the student's intuition.
(2) The book uses modern techniques when applicable. Just because this book teaches the material in a concrete/classical setting does not mean that its methods are outdated. The student will become very used to modern techniques, but applied here in easier settings than what you would find in a standard graduate leveled book. Hence, when the student eventually takes graduate leveled courses, he or she will come to see the definitions and techniques as natural extensions of those learned previously.
(3) The student learns the classical theory first, which entirely motivates the modern theory. Going back to (2), this allows students to view the modern theory as a natural extension of the classical theory, and hence their intuition learned from this book is still applicable in the more abstract settings. Also, for arbitrary manifolds, our intuition comes entirely from surfaces in R^3. The fact that this book has a 100+ page chapter on surfaces alone makes it worth reading.
The material covered in this book is expansive, and i think students will find that even in more advanced Riemannian geometry courses they will see material from this book still being taught. Hence, it will be a long time before a student completely outgrows this book.
I've said only good things, but there is one thing that annoys me. This book drops the ball in providing intuition in several topics. For example, the book defines Christofell symbols as abstract sums of inner products. However, the point is that they provide the scalars for the coordinates of mixed partial derivatives in a tangent space. There is a theorem that implies this, but this intuition is never explicitly stated. In a book aimed at undergraduates, I feel that everything, including intuition, should be stated and highlighted. This has not been too bad of a hinderance for me, because a quick trip to the professor's office will often sort things out for me. However, any self-learners may get the wrong idea about how to thing about certain topics. This is not too bad of a problem, and for the most part this book does a good job at explaining and motivating topics, so I do not feel that any more than one star should be taken off.
Prerequisites for reading:
(1) Strong Linear Algebra background. And by this I do not mean computations in Lay, I mean theory in Axler or something comparable. Often times, change of bases are applied without being explicitly stated in the proofs, and you should be able to pick up on this immediately.
(2) Analysis-both single variable and multiple variable. You do not need anything past the basics of multivariable differentiation and integration, i.e. no Stoke's theorem or differential forms are needed.
(3) While not completely necessary, there are a few proofs which use the existence and uniqueness theorem of ordinary differential equations, so knowing this exhausts all possible prerequisites. I do no know ODE theory, and I am not having trouble understanding the book as a whole, so this prerequisite may be relaxed/forgotten.