I have learned about curvature of curves given as graphs of vector functions $\vec{r}(t)$ in three dimensions. I understand that the curvature is a measure for how much the curve is bending. I know that the curvature can be calculated using this simple formula $$ \kappa (t) = \frac{\lvert \vec{r}'(t) \times \vec{r} '' (t)\rvert}{\lvert \vec{r}'(t)\rvert^3} $$
I had heard that the concept of curvature can be extended to surfaces and even spaces. For this question I would like to know how one can calculate the curvature of a surface given by, for example, and equation like $f(x,y,z) = 0$. If this is too broad a question, then I would be interested in just seeing how one can calculate the curvature of a sphere and a plane.
The only thoughts I have on this is that the curvature should be the derivative of a normal vector with respect to some kind of arc length. But I don't understand the equivalent definition of arc length for a surface.
If this is still too much, then I would be happy to have a reference that a average student in calculus could understand.
There is more than one notion curvature for surfaces. Let's consider a surface $M\subseteq \Bbb R^3$. We can get a unit normal vector in any point, and so we have the Gauss map $N: M \to \Bbb S^2$. You can define the Gaussian Curvature by $K(p) = \det(-dN_p)$ and the Mean Curvature by $H(p)= \frac{1}{2}{\rm tr}(-dN_p)$. One can prove that $-dN_p$ is diagonalizable, so we have eigenvalues, called the principal curvatures. In more advanced contexts one deals with the Riemann-Christoffel curvature tensor, the Ricci tensor, scalar curvature, etc.
We don't have a notion of arc-length for surfaces, so we deal with total derivatives (differentials, the derivative as a linear map).
The question is very broad and there is a whole theory about these things. I recommend taking a look in books on Differential Geometry, such as John Oprea's Differential Geometry and Its Apppications, and Barret O'Neill's Elementary Differential Geometry, for example.