To my mind, there seem to be two main paths to differential geometry. There is the classical approach, focusing on curves and surfaces in $\mathbb{R}^n$, especially $\mathbb{R}^3$. Prerequisites tend to be multivariable calculus and linear algebra. Some exposure to real or complex analysis and metric spaces or point-set topology is helpful, but not strictly required. A couple of titles following this approach include Pressley's Elementary Differential Geometry and Tapp's recent Differential Geometry of Curves and Surfaces.
The modern approach defers much of the geometry and studies manifolds first. An example of this path would be Lee's trilogy: Introduction to Topological Manifolds, Introduction to Smooth Manifolds and Riemannian Manifolds: An Introduction to Curvature.
Consider a scenario in which upper-division undergraduates study classical differential geometry and then return to the subject at graduate level, the latter with a manifolds-first approach. Would the classical introduction help or hinder?