Let $R$ be a commutative (unitary) ring. Every $R$-module is the quotient of a free $R$-module, and every free $R$-module with basis $B$ is isomorphic to $R^{\oplus B}:=\bigoplus_{i\in B}R$. That is, every $R$-module can be written as a quotient
$$R^{\oplus I}/N,$$
where $I$ is some index set and $N$ is a submodule of $R^{\oplus I}$. I'm looking for ways to categorize these quotient modules in order to categorize all $R$-modules. One that comes to mind immediately is the fundamental theorem on finitely generated modules over principal ideal domains: Every finitely generated module over a principal ideal domain $R$ is of the form
$$\bigoplus_{i=1}^n R/(a_i),$$
where $a_i\in R$ and $a_i\vert a_{i+1}$ for $i=1,\dots,n-1$. This also means that for PIDs, $R^n/N\cong\bigoplus_{i=1}^n Q_i$, where $Q_i$ are quotients of $R$, which is the kind of thing I'm looking for. Though this is a non-trivial theorem and it requires strong conditions: a finite generating set and we need a PID. Are there more elementary results about the structure of quotients of free modules?
Edit: To clarify, I'm not looking for a complete classification of all possible modules. Partial results, for instance ones holding only over certain kinds of rings, are also fine. Im mostly looking for really broad categories of modules, not the kind of complete categorization we have for vector spaces.