In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in general coordinate systems. This appealed to me because I wanted to understand this from the view of differential geometry, instead of the long, ad hoc computations for cylindrical and spherical coordinates. I haven't read Spivak's book, but have I have a little understanding of basic differential geometry (first two chapters of Tu's Intro to Manifolds). Anyway, there's an online version of Spivak's text, but looking through the table of contents, I'm not really sure where I'd find this treatment of differential operators in curvilinear coordinates in his book. If anyone's familiar with it, could you please cite where in the book he treats this topic? Or if anyone is familiar with another reference or could provide a good explanation, that would also be greatly appreciated.
I'm not completely sure what you want, but honestly the entirety of Spivak's Calculus on manifolds is devoted to exactly that. If you want something that feels familiar, you can simply find $\nabla$ in various coordinate systems in Wikipedia, but if you want a less coordinate-centric view then you're probably going to need to step outside of your comfort zone.
In particular, you should be aware that the concepts of grad, curl and div are not particularly useful by themselves in an arbitrary manifold with arbitrary coordinates. Instead, functions and vector fields are replaced by differential forms, and div/grad/curl get replaced by the exterior derivative $d$.
With that in mind, the sections of Calculus on manifolds which deal most directly with these things are §4.2 (Fields and forms), §5.2 (Fields and forms on manifolds) and §5.3 (Stokes' theorem on manifolds). Honestly, though, you should read the whole thing: it's very short (137 pages of large type on a small page), and it's delightfully constructed. It's honestly a little jewel! (On the other hand, the compact construction does make it a bit dense at times.)
Beyond this, if you want a more expanded view of what happens to functions, vector fields, and differential operators on more general manifolds, I would really recommend Spivak's A comprehensive introduction to differential geometry (vol I). In particular, §3.4 (The tangent bundle of a manifold) and chapters 7 (Differential forms) and 8 (Integration) deal with (the appropriate generalizations of) $\nabla$ from a general differential geometric perspective.
I would say that the Wikipedia page on curvilinear coordinates and the article Mathematical Physics Lessons - Gradient, Divergence and Curl in Curvilinear Coordinates by James Foadi are enough to understand what is going on.