Differential geometry in the context of manifolds

I am an undergraduate student of mathematics. I have a solid background on calculus, linear algebra, real analysis and point set topology, but I have never studied differential geometry.

I am very interested in studying differentiable manifolds, but, should I first study classical differential geometry? Should I first study curves, surfaces and curvature? I make an emphasis in the word 'should', because I've been told it is not strictly necessary to master these concepts.

I think it's important to know first how deeply you want to study differential geometry/differentiable manifolds. I agree completely with Mike Miller's comment above, but would like to add a few thoughts. There is a big distinction between just studying differentiable manifolds and differential geometry. Geometry relies entirely on the definition of smooth manifolds, but has an added structure called a Riemannian metric (which is basically an inner product defined at each tangent space, but which varies smoothly as we go from point to point on a manifold). One might think you need to master the study of manifolds before studying geometry, but this isn't necessarily so: there are lots of topics in manifold theory which are not prerequisites to studying geometry.

Without a Riemannian metric we can still study vector fields, covector fields, tensors, calculus, topology, Lie groups and Lie algebras. All of these are exciting topics. With a metric, we can however add to that list connections, geodesics, curvature, and applications to physics. Either route you take, you will need to understand some amount of manifold theory. The question is: how much do you want to know?

If this is an area of math you would like to just dabble in, then I concur with Muaddib's recommendation of Do Carmo's Differential Geometry of Curves and Surfaces which deals entirely with 2-dimensional manifolds (i.e. mostly objects you can visualize) and adapts most of the language and machinery of differential geometry to the 2-dimensional setting. Either way, this is a pretty good book to get started with. Do Carmo has a more advanced book Riemannian Geometry which is very popular, although the text moves rather quickly and especially at the beginning (make sure you are well-versed in multivariate analysis before starting this text).

If on the other hand you think you are a geometer at heart, and maybe want to specialize in manifolds in grad school, then I would recommend John Lee's trilogy of books: Topological Manifolds, Smooth Manifolds, and lastly Riemannian Manifolds. This is certainly a much slower route to take, but if this area of math is your passion then it's certainly worth the investment. Lee's books are very pedagogically written, self-contained, and pretty much written such that if you go in his sequence you can start with a very small amount of prerequisite knowledge (literally just analysis, linear algebra, group theory, and maturity). It's an added bonus that he's very active on this site and can answer questions if you get stuck. As a side note from my own personal experience, I bought a first edition of the second book (smooth manifolds) and found out that Lee made some drastic alterations between the first and second editions. My guess is that the newer edition is a little bit easier to work through than the first.

There are many other classic texts (most of which I probably don't know). Some of the other texts that exist are Spivak's Calculus on Manifolds, and Munkres' Analysis on Manifolds. These two books cover a lot of the same material, but Munkres is slow while Spivak is quick. Additionally they only discuss manifolds in the context of generalizing calculus to non-Euclidean spaces (there is much more to manifolds/geometry than just calculus). Warner's book on Manifolds and Lie Groups is also classic and offers a blend of advanced topics in smooth manifold theory and has the advantage of being concise (no geometry here though).