I think there are two ways to approach a first course on manifolds: one can focus on either their geometry or their topology.

If you want to focus on geometry, then I think Anton Petrunin's suggestion is the end of the story. I'm a fourth year graduate student, and practically every time I find myself confused about something in differential geometry I realize that the root cause of my confusion is that I never properly learned surfaces. And I've taken lots of geometry courses.

If you want to focus on topology, I really think it makes a lot of sense to teach some Morse theory. It's rather elementary, it's extremely powerful and virtually ubiquitous in differential topology, and most of all it really feels like topology in a way that differential forms don't.

Finally, from looking at only the two books you mentioned in your question, I would be a little worried that your students won't have a lot of examples to work with. What about introducing Lie groups?