What should be taught in a 1st course on smooth manifolds?

I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness these lecture notes by Peter Petersen, where it is proved in a few lines on page 9, the prerequisites being about two pages long.

Bjørn Ian Dundas and our friend Andrew Stacey also have online documents proving this theorem.

If this is the first course in Differential geometry, you should not go further than Gauss--Bonnet for surfaces. I would not even consider anything with dimension >2. By the way here is our textbook on the subject. If they like Differential geometry, they could take another course.

If you cover more, then it is easy to produce lammers. If you skip these topics, then (most likely) your student will have no idea what is differential geometry at the end of the course.

The problem will be that the students do not have a firm grasp of multivariable calculus.

You should probably start with a rigorous review of multivariable calculus including the definition of the differentiable, C^1 implies differentiable on open sets, mixed partials are equal, inverse function theorem, local immersion theorem, local submersion theorem. That will allow you to segue into the definition of smooth manifold as a parametrized subset of R^n as in Guilleman and Pollack.

Guillemann and Pollack is a softening of Milnor's "Topology from a Differentiable Viewpoint" and as such is about the lowest level approach you can take to introducing the students to the "stuff" of topology. The exercises are good. I like to have the students divide up the long guided exercise sections to present at the board. I like to supplement the book by proving the Morse Lemma, having a discussion of linking number, and proving the the Hopf fibration is not homotopic to a constant map using linking numbers. I also like touching on complex variables by proving the argument principle. Finally, I like proving that two maps from a closed oriented n-manifold to the n-sphere are homotopic if and only if they have the same degree. I don't do all of these in any one year as there is not time. I generally key off of what seems to interest the particular group of students in the class that year.

Be careful in the section on integration, they leave out (or left out in an earlier edition) that you need to be using orientation preserving parametrizations to define the integral.

After teaching such a course for about 15 years, I changed directions and started teaching the foundations of smooth manifolds in the place of the Guilleman and Pollack course, so that students could learn a more mature definition of smooth manifold, and introduce vector bundles, tensors, and Lie Groups. I have used both the books by Jack Lee and by Boothby. Each has its strong points and weak points (at least in use with graduate students at Iowa.) This turned out to be better for the graduate program as a whole because kids who wanted to do representation theory or PDE could get exposed to the ideas they would see in their research. It also allowed the Differential Geometry sequence to run more regularly. If you decided to go that route, it would still be wise to start with multivariable calculus, as really, very few kids going to graduate school in math have a sufficient background in the calculus.

However, the students are much less happy about taking the foundations of smooth manifolds, because it does not offer the immediate gratification of studying degree and winding number. In fact, when I teach the course as foundations of smooth manifolds, there will always be a block of 3 or 4 students who resent having taken the class. When I teach out of Guilleman and Pollack, even the students who never develop a clue, still enjoy the experience.