I am trying to understand if there is a conventional way to read super- and subscript notation of metric tensors. Is there a canonical way of doing this?
For instance, what is the difference between $R^{\mu\nu}$ and $R_{\mu\nu}$? Is this difference purely contextual, or do the subscripts carry intrinsic meaning? How do you deal with combination super- and subscript tensors, e.g. $R^\mu_{\alpha\beta}$?
Since your question was about the metric tensor, I'll use that as my example.
To answer your question, $g_{ij}$ is a $function$ defined in a neighborhood of your manifold. More specifically, since $g$ is a $(0,2)$-tensor field, we can write $g = \sum_{i,j} g_{ij}\hspace{2pt} dx^i \otimes dx^j$. That is, in a neighborhood, we can expand $g$ in the basis $\{dx^k\otimes dx^\ell\}_{k,\ell}$, where $g_{ij}$ are the coefficient functions. Some linear algebra tells us that $g_{ij} = g(\partial_i, \partial_j)$. If you think about $g$ as a matrix whose $ij$ entry is $g_{ij}$, then $g^{ij}$ is the $ij$ entry of the inverse matrix.
Slightly more generally, if you have an $(1,3)$-tensor $A$, then $A^i_{jk\ell}$ is a function defined in a neighborhood of your manifold by $A(dx^i,\partial_j,\partial_k,\partial_{\ell})$.
