soft question - Why is it a good idea to study a ring by studying its modules?

I want to answer your question twice: first with a "top-down" approach and second with a "bottom-up" approach. Let me limit myself to the first answer here and see how I do.

I claim the following analogy:

abstract groups : group actions on sets :: abstract rings : linear actions of rings on
abelian groups (= modules)

I will take it as mostly self-evident that it is desirable to study groups acting on sets. If you are not doing this -- but thinking of groups only as sets with a certain law of composition -- then you are thinking about groups "in the wrong way". You are missing out not only on powerful tools for studying abstract groups (e.g. the Monster was constructed as the automorphism group of a certain algebra), but also, even more importantly, on why groups are important and interesting to mathematics: they come up as automorphisms of things, not (or rather, rarely) abstractly.

The way to think about a group action is that you have a set S, and it has an automorphism group, Sym(S), the group of all bijections from S to itself. Then an action of G on S is simply a homomorphism of groups G -> Sym(S). More generally, if x is any object in a category C, then it has an automorphism group Aut(x), and one can think of a homomorphism from an abstract group G to Aut(x) as a group action on x.

Now in place of a set, we take an abelian group M. This has more structure -- apart from a group Aut(M) of Z-linear automorphsims, it also has an endomorphism ring End(M): the ring of Z-linear maps from M to itself. Note that End(M) is in general noncommutative, so this construction is more general than any "ring of functions" construction in (commutative!) algebraic geometry.

So given a ring R, the analogy is completed by considering ring homomorphisms R -> End(M). As for rings, this provides a bridge between the abstract notion of a ring and the "real world" notion of endomorphisms of an abelian group. Moreover, just as the notion of a symmetry group of a set generalizes to that of an automorphism group of an object in a category, similarly, any object x in an abelian category C has an endomorphism ring, and hence one can consider "ring actions" of an abstract ring R on x, via homomorphisms R -> End(x).

In the first part of the analogy, a distinguished role is played by the category of [left, or right] G-sets for a particular group G. In particular, this provides a way for every group G to be the precise automorphism group of some object in a category: it is the automorphism group of itself. It is of course not true that any abstract group is equal
to the full symmetry group Sym(S) of a set S, so this is an important construction.

In the second part of the analogy, a distinguished role is played by the abelian category of [left, or right] R-modules. In particular, End_{R-Mod}(R) = R, so every ring is the precise endomorphism ring of some object in an abelian category. (I am pretty sure that not every ring is isomorphic to the full endomorphism ring of an abelian group, although this is less obvious than the other case. It might make a good question in its own right...)

This is I think the right "general" answer to the question "Why is studying modules of a ring a good way to understand that ring?" A different kind of answer would give instances in commutative algebra when theorems about rings are proved using modules. I'll try that at some future point.