More recent introductory text on Differential Geometry similar to Kobayashi/Nomizu We began an introductory course on Differential Geometry this semester but the text we are using is Kobayashi/Nomizu, which I'm finding to be a little too advanced for an undergraduate introductory course in DG. There are also no graded homeworks, quizzes, or exams so a text with solved problems would be preferred.
Textbook recommendations for introductory DG books is not a new question here, but I was specifically looking for books that follow a similar formalism as Kobayashi/Nomizu.
 A: I second some of the other recommendations (Tu and Hamilton's books both seem very good from my quick look at them). Another option is the two books by Gregory Naber: Topology, Geometry and Gauge Fields: Foundations and Topology, Geometry and Gauge Fields: Interactions. They're both very clear, extremely explicit in their proofs and calculations, and at least make an attempt to have some exercises for the reader.
A: Two (bright new) interesting books by Jean Gallier and Jocelyn Quaintance:

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*Differential Geometry and Lie Groups: A Computational Perspective.

*Differential Geometry and Lie Groups: A second course.

A nice introduction:

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*First Steps in Differential Geometry: Riemannian, Contact, Symplectic, by Andrew McInerney

*Differential Geometry: Curves - Surfaces - Manifolds, by Wolfgang Kühnel

Applied to physics:

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*Differential Geometry and Relativity Theory: An Introduction, by Richard L. Faber

*General Relativity Without Calculus: A Concise Introduction to the Geometry of Relativity, by Jose Natario

*Tensors, Differential Forms, and Variational Principles, by David Lovelock & Hanno Rund

and many, many more...

Update 1:
After the comment by @BenMcKay, I found a set of (not as well-known as they should) lectures by Nomizu himself, published by the Mathematical Society of Japan in 1956. These are titled: Lie Groups and Differential Geometry.
Just for your information, these notes are about 80 pages long, and has three chapters:

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*Differential manifolds.

*Connections in fibre bundles.

*Linear connections.

Another useful reference is:

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*Analysis, manifolds and physics Part I: Basics. by Y. Choquet-Buhat, C. DeWitt-Morett and M. Dillard-Bleick.

There is a Part II of this book, by the first two authors, focus in the applications.
A: Yikes, that's brutal - Kobayashi-Nomizu is an excellent reference text, but using it in a first course on the subject is a bit like learning English from the Oxford English Dictionary.  For instance: chapter 2 is about connections on principal bundles, chapter 3 is about linear / affine connections, and chapter 4  is about the special case of Riemannian connections; this is conceptually an elegant way to build the theory, but pedagogically it is exactly backwards.
I second the suggestions in the comments to at least start with curves and surfaces.  If the course has to go beyond that then it gets tough - there are good books about curves and surfaces, and there are good books about connections on vector bundles, but there aren't many that do both subjects in a unified way.  In fact the only example that I know is Loring Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes, which covers both branches of the subject and bridges the gap with explicit calculations involving Riemannian connections on surfaces in $\mathbb{R}^3$.  It has a modest number of problems at the end of each chapter, and they're generally pretty good if not numerous.
