I understand the different types of quotient objects, but I still struggle with them - which I think is a result of me neglecting to work on them despite knowing it's a weak area of mine. How I generally learn about a concept in math is by solving more difficult problems about or using that concept. So my question is simply this.
Are there any good exercise lists for difficult problems dealing with quotient groups, rings, spaces, modules, sets, algebras or lie groups, etc, etc.
Let me propose a way of thinking about quotients a bit differently and you can distill some exercises from this post.
Let's stick to the group case, although much of these ideas extend beyond groups. Given a group $G$ and a normal subgroup $H\subseteq G$ one can define the quotient group $G/H$. As a set $G/H$ is just the collection of left $H$-cosests. The group operation is then simply defined by $gH\cdot g'H=gg'H$. You can verify that $H$ needs to be a normal subgroup for $G/H$ to be a group.
Okay, that's straightforward enough although it might not immediately be clear why this is important so let's view things differently. Given a surjective morphism $f\colon G\to C$ of groups, one has that $\ker(f)$ is a normal subgroup of $G$ and $C\cong G/\ker(f)$. So quotients really just boil down to surjective maps.
Now look at surjection $f\colon G\to C$ and think about $C$. Clearly $C$ can be generated by some collection of elements of the form $f(g)$ and relations that hold in $G$ can be pushed to $C$. For example if $g^2=1$ for some $g$, then $f(g)^2=f(g^2)=1$ in $C$ as well. Of course $f$ can induce even more relations, for example $f(g)$ could be $1$ to begin with which is a stronger relation than $f(g)^2=1$. These additional relations are controlled by $\ker(f)$. So now we have a very powerful way of thinking about $C$. You can think of $C$ as just $G$ with a bunch of additional relations which are determined by $\ker(f)$. In a way, you can think of $C$ as a dumbed-down version of $G$ which, depending on how much information you killed, still contains some information on $G$. (Edit: See presentations of groups).
Often a given group is too complicated to study directly, so one can impose extra relations and try to distill information from this quotient. For example, given a group $G$, the center $Z(G)$ is a normal subgroup. If $G/Z(G)$ is cyclic, then $G$ is commutative. As a corollary one finds that any group of order $p^2$ (with $p$ prime) is abelian. So here general nonsense on quotients yields a very concrete tool for studying groups.
Now the amount of useful exercises and applications on just quotients is somewhat limited but the idea of breaking up a structure into easier parts can be pushed further. I suggest you look at solvable groups and nilpotent groups were one considers a group $G$ together with a chain $$\{1\}=G_0\subseteq G_1\subseteq \dots \subseteq G_n=G$$ satisfying several conditions on the quotients $G_{i+1}/G_i$. There are many good exercises on these type of groups (for example: If $N\subseteq G$ is a normal subgroup, then $G$ is solvable if and only if both $N$ and $G/N$ are solvable). Solvable and nilpotent Lie algebra are defined in a similar manner and satisfy similar properties.
If you are capable of proving these types of result with ease, I think you are fairly comfortable working with quotients.