To start, it's worth being a bit clearer on where ideals come from. To quotient a set, we need an equivalence relation on that set. If that set is equipped with algebraic operations, e.g. it is a ring, then for the result to be the same sort of algebraic structure, the equivalence relation needs to respect the algebraic operations. An equivalence relation that respects the relevant algebraic operations is called a congruence. Given some congruence, $\approx$, on a ring, $R$, we can form the quotient ring $R/{\approx}$ simply by quotienting the underlying set. The original operations will be well-defined because $\approx$ is a congruence (and not just an equivalence relation).
Because $\approx$ is a congruence, we have that $r\approx s\iff r-s\approx 0$. The equivalence class of $0$, i.e. $[0]=\{r\in R\mid r\approx 0\}$ is exactly an ideal. That is, every congruence gives rise to an ideal by considering the equivalence class of $0$, and every ideal, $I$, gives rise to a congruence via $r\sim s\iff r-s\in I$. This is related to the first paragraph of Bill Dubuque's answer. This is also exactly the same story for normal subgroups. For modules, it just so happens1 that the analogue of an ideal is itself a submodule. So it's not that for modules we care about substructures and for rings we care about ideals. We care about the same thing in both cases, it just so happens that the "ideals" of modules coincide with submodules.
Before I continue, one interesting fact is that subobjects and quotient objects are actually dual (in the categorical sense) notions (at least in regular categories which includes any category of algebraic structures, such as the category of rings).
As indicated in Zev Chonoles answer, subobjects are far from unimportant. Nevertheless, ideals take a lot more of the spotlight. I'll give two reasons why that may be the case. I'll start with a more sociological answer, and then I'll give a more technical answer.
The sociological answer is simply that ideals are a more challenging to understand concept than subrings, so more time is spent on them. For any algebraic object (for a technical sense of algebraic, but one that captures most examples with the notable exception of fields), a subobject is exactly the thing you think it would be. Namely, a subset for which the original operations are closed. Quotient sets in general are harder to understand than subsets, and ideals are often presented in a way that pulls the definition out of a hat. Even when the connections to congruences are made clear, it is still fairly complicated. The notion is also not as uniform across algebraic structures. While quotienting by a congruence is something that is meaningful for any algebraic object, there is not always a distinguished constant to consider the equivalence class of like $0$, and even if there is it isn't always helpful. In the examples above, all relied on the group structure of the algebraic structures.
Early examples are also misleading. The "ideals" of vector spaces (and more generally modules) are exactly sub-vector-spaces. For groups, they are normal subgroups. These examples make it look like we should be quotienting by substructures. There's also an aesthetic niceness of being able to "divide" vector spaces. The ring case looks out of place since an ideal isn't a subring, not even a special type of subring, and thus we can't "divide" rings. This likely leads some to think that a ring with identity is the wrong notion, and we should be considering rings without identity, often called rngs, because ideals are subrngs.
On the more technical side, every algebraic structure is a quotient of a free structure. This is almost the defining property of being "algebraic". This is, of course, the basis of the notion of a presentation. From this perspective, we care more about ideals because they to congruences and thus quotients, and quotients are more fundamental to algebraic structure than subobjects.
Another fact is that while categorical limits (which include subobjects and products) are given in a simple way for any algebraic category, colimits, which will typically involve quotients and which quotients are an example, need not be. For example, the (categorical) product of two rings is just defined by the (categorical, i.e. cartesian) product of their underlying sets with the operations defined component-wise. This works for any algebraic structure, e.g. the (direct) product of groups or products of lattices. The categorical coproduct, on the other hand, while it always exists for algebraic categories, is not so simple. The coproduct of two groups or two rings is not at all the coproduct (i.e. disjoint union) of their underlying sets. Indeed, the easiest approach is to consider free objects, where the answer is easy, and then quotient that, hence the "free product" of groups which is the coproduct. The upshot of this is that a lot of the time colimits are going to be where interesting things happen and where different algebraic structures really show different behavior. For example, for abelian groups and modules finite products and finite coproducts coincide, but this is certainly not the case for (arbitrary) groups and rings.
To be clear, a lot of ring theory is secretly module theory. Similarly, a lot of group theory is secretly the theory of $G$-sets. Groups are to rings as $G$-sets are to modules. Many results are easier and more intuitive when you view them as results about modules which can then be specialized to rings. Categories of modules are also abelian categories, unlike the category of rings, which provides a lot of nice laws and structure which simplifies things immensely.
1 This is not completely an accident and follows from the fact that categories of modules are abelian.
