Learning modern differential geometry before curves and surfaces

Solution 1:

You won't miss much by dropping curves and surfaces: every important article I studied, browsed or heard about published in the last 60 years in differential geometry by such luminaries as Thom, Milnor, Atiyah, Hirzebruch, Perelman,...contains little or no reference to curves and surfaces.
On the other hand if you spend your time on Codazzi equations, Frenet-Serret frames and umbilic points you might have no time left for principal bundles, Stiefel-Whitney or Chern classes, cobordism,etc. and that means you will have little chance of understanding anything in modern differential geometry.
Of course it would be great to combine the mastery of both the exquisitely detailed classical results in one or two dimensions and the general powerful modern techniques of differential geometry/topology, but if you want to arrive at the frontier of research in a reasonable time you will have to favour the latter over the former.

Solution 2:

I agree that you'll be fine if you go straight into manifolds before curves and surfaces. But it is simply NOT true (as stated in another answer) that surface is of little interest in modern different geometry, in particular in area related to PDE and analysis.

In case of curves, that is indeed not so interesting as all 1 dimensional objects are locally isometric (So there are essentially no intrinsic properties to study). But there are still something extrinsic to talk about (for examples geodesic in Riemannian manifolds and closed orbit in symplectic manifolds). The studies are quite complete though (So really not so interesting from the research point of view).

But that is not the same for dimensional two. The main reason is that the most natural operator, the Laplace operator, is conformal invariant when dimensional is two. This results in a large contrast between surface theory and general manifold theory. Indeed, there has been large progress in minimal surface theory, mean curvature flow of surfaces, when compared to that in general dimension.

On the other hand, the use of surface theory is extremely essential in studying manifolds of positive curvatures, puesdo-holomorphic curves in symplectic manifolds etc, which are all active research directions in modern differential geometry.

Similar special phenomenon occurs when you restrict to 3 and 4 dimensional manifolds. These "low dimensional geometry" are also very active research directions.

However, I do agree that you can put aside the curve and surface theory for the moment. In a limit amount of time, it is reasonable to go into the general pictures first. I just feel like I have to say something for surface.