I support the previous suggestions, especially Ted Shifrin's fine notes.
I am quite ignorant of differential geometry myself, hence am sympathetic to this proposal. I have recently realized, through my love of Euclidean geometry, what others probably all know, that there is a natural sequence of topics: euclidean geometry, spherical geometry, hyperbolic geometry (these all being surfaces of constant curvature), and then other surfaces of constant curvature, namely quotients of the previous ones.
Next one naturally progresses to surfaces of varying curvature, i.e. the realm of differential geometry proper, Riemannian surfaces. Finally one raises the dimension.
Thus the natural elementary course to teach seems to me to be spherical geometry, then....
The point is to teach curvature, and emphasize that euclidean geometry is the unique geometry of constant zero curvature, (with another hypothesis that lines are infinitely long).
In particular we seem to miss an opportunity when we teach non euclidean, i.e. hyperbolic geometry, without a link to differential geometry via curvature, but this is normally done in non euclidean geometry courses.
E.g. curvature is easily presented in the elementary way that Riemann (and Gauss) described it, as the defect of the angle sum of an infinitesimal triangle. Gauss Bonnet easily follows for the basic surfaces. In this regard, I suggest a first step in learning differential geometry is to discuss whether a cylinder is or is not curved, and why or why not.
Along this line of reasoning, John Stillwell has a very nice book on surfaces of constant curvature that would impart a lot of useful concepts at the advanced undergraduate levellevel..
I even sketched out a plan form such a course to present to brilliant 10 year olds, assuming neither calculus, topology, linear algebra or even trig, which could be taught in the course
For more general differential geometry, there is a book by David Henderson, which attempts to teach an intuitive understanding of the ideas, curvature, parallel transport, holonomy...
But the basis of this suggestion is the hypothesis that concepts are more fundamental than techniques for computing them. Since it seems that the most important concept in differential geometry is curvature, the first job is to convey an appreciation for curvature and its role in geometry. This can be done naturally in a very elementary setting. Only afterwards does it seem important to train someone in the means of computing it, i.e. tensor calculus and differential forms, chern classes, etc....