TITLE: More recent introductory text on Differential Geometry similar to Kobayashi/Nomizu
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: Two (bright new) interesting books by Jean Gallier and Jocelyn Quaintance:

Differential Geometry and Lie Groups: A Computational Perspective.
Differential Geometry and Lie Groups: A second course.

A nice introduction:

First Steps in Differential Geometry: Riemannian, Contact, Symplectic, by Andrew McInerney
Differential Geometry: Curves - Surfaces - Manifolds, by Wolfgang Kühnel

Applied to physics:

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:

Differential manifolds.
Connections in fibre bundles.
Linear connections.

Another useful reference is:

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.