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.