good point. think about the goal of differential geometry, to understand curves and curved surfaces. the basic problem is how to emasure the curvature.
This is done by comparing the curves and surfaces you have with the simplest curved objects, namely circles and spheres.
Essentially a curve (which is not a circle) has the same curvature at a point as a circle tangent to it at the point and which also crosses it there. the curvature of a circle is measured by its radius, or ratehr the reciprocal of the radius, since that gets larger as the circle gets tighter.
another way to compoare the curve to a circle, if the curve is in the plane, is to send each point of the curve to the end point of the unit vector at the origin, which is parallel to the normal vector to the given curve at each point.
We also have a choice of length of the vector, instead of a unit vector. If we choose the length of the normal vectors to map to, so that the derivative of the map is one wrt arc lnegth, at our point, then we are mapping to the circle with the same curvature as our curve.
This method of comparing curvature by compoaring arclength generaliuzes to surfaces, as Gauss realized.a surface in three space is compared with a sphere centered at the origin. we simply map each point of the surface to the endpoint of a vector of fixed length, emanating (I love that word) from the origin, and parallel to the normal vector to the surface at the point.
then we compare areas under this map, by taking the determinant of the derivative (called the jacobian determinant). The sphere has the same curvature as the surface at a given point, when the area change is one, i.e. the area determinant is one. This happens when the radius vector from the origin has appropriate length.
the curvature is apparently measured by the square of the reciprocal of the curvature of a geodesic circle on this sphere, but I am no tpositive since I ma making all this up. I.e. the usual definition of Gauss curvature is the valoue at the point of the jacobioan determinant, maybe divided by pi or somehting.after all this is my book. but see how simple it all is? the crap in diff geom books with huge arrays of indices and coordinates is just an attempt to assign numbers to these ideas. What does it profit a man to learn to manipulate all those indices and lose his soul? no wait that's from the hebrew bible - I meant: what good is it to throw around words like connection, and chern class, or contravariant, or lie derivative, and not understand the meaning of curvature?
Spivak starts you out just like this: even as we take a limit of two points to define a tangent line, so dow e take a limit of three points to define a circle of curvature. then he goes staright to gauss, and finalkly tyranslates it for you into fancy notation but with a conceptual underpinning.
he introduces all the may versions opf modern curvature, and checks each one in the "test case". Namely he checks that each one gives zero for flat euclidean space. then he shows the acid test, that zero curvature implies locally flat in the sense of locally equivalent (isometric) to euclidean space. at least as i recall. it has been 30 years since i read spivak, and i only spent 2 days reading it (vol 2).
and he gives you many extras, like how to compute the dimension of a lie group, how to work with de rham cohomology, how to drive a significant global result from a trivial local one (sheaf theory)...
It is good to have a familiarity with 3-dimensional differential geometry first (i.e. geometry of curves and surfaces embedded in R^3). However you need to sort of re-learn from an intrinsic point of view when you go to Riemannian geometry or GR for that matter. So just be prepared. 