How do ideas in differential geometry expand upon ideas from introductory calculus I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that of a function that I am now used to. Furthermore the notion of a vector field seems to be much more nuanced.
Can someone elaborate on how some simple ideas from introductory calculus are expanded upon through the study of differential manifold? 
For example, what is the equivalent of a manifold in introductory calculus? Are charts the same concept as functions? Are vector fields differential operators same as $\partial$?
 A: The fact is that the objects of study in differential geometry (smooth manifolds) build on and generalize the objects of study in elementary geometry (lines, circles, spheres, cylinders, etc.) much more than they generalize the objects of calculus.  It's the tools of calculus that are directly generalized to create the fundamental tools of differential geometry. Differentiation of functions gets generalized to create such concepts as tangent vectors to manifolds, velocity vectors to curves, etc. Integration of functions gets generalized to line integrals of differential forms, volumes of manifolds, average values of functions.  These are just the tip of the iceberg. From there, the path leads into sophisticated advanced concepts like de Rham cohomology, Lie groups, connections, and symplectic structures, none of which would make any sense without the foundational tools provided by calculus.
A: This question perhaps risks closure on account of the (in my view) enormous breadth of the main question, but I'll try to answer the specific questions asked afterward. I'll assume "introductory calculus" simply means single- and multivariable differential and integral calculus.
An analogue of a differentiable manifold is (very roughly) anything one integrates over: an interval $(a, b) \subset \Bbb R$, a curve in $\Bbb R^2$ or $\Bbb R^3$, a region in $\Bbb R^2$ or $\Bbb R^3$, a surface in $\Bbb R^3$, etc. (One could also take the analogue to be, even more roughly, "anything that admits a tangent vector field", though it's perhaps less intuitively clear then why $(a, b)$ belongs on this list.)
(Smooth) charts are not the same as functions, as they must always be diffeomorphisms, and in particular homeomorphisms, whereas smooth functions need not even be bijections. One should think of charts as the inverses of differomorphic parameterizations, e.g., of curves and surfaces; just as some surfaces (e.g., the $2$-sphere) admit no parameterizations that are diffeomorphisms, i.e., that cover the whole surface at once in a nice way, in general manifolds cannot be covered with a single chart.
And yes, vector fields are roughly an analogue for the differential operators $\partial_{x^i}$ on $\Bbb R^n$. But, vector fields in multivariable calculus are just a special case of vectors fields on differentiable manifolds, and any choice of chart on a differentiable manifold determines coordinate vector fields on the domain of that chart (and in general, changing the chart changes those vector fields). Also, Clairaut's Theorem assers that mixed partial derivatives of smooth functions $f$ commute, that is, e.g., $\partial_x \partial_y f = \partial_y \partial_x f$, but in general vector fields do not; rather, for any two vector fields $X, Y$, we get a new vector field $[X, Y]$ (called the Lie bracket of $X$ and $Y$) uniquely characterized by the formula $X \cdot (Y \cdot f) - Y \cdot (X \cdot f) = [X, Y] \cdot f$.
You didn't ask about this specifically, but since it's fundamental, especially in light of the above description that a differentiable manifold is something one integrates over, I'll point out that Stokes' Theorem, usually written as
$$\int_{\partial M} \omega = \int_M d\omega$$
generalizes all of the familiar "Fundamental Theorems of Calculus" from single- and multivariable calculus, including the single-variable F.T.C., the F.T.C. for line integrals, Green's Theorem, Gauss' Theorem (a.k.a. the Divergence Theorem), and (the classical) Stokes' Theorem.
