Quotienting a group by an equivalence relation

Sure. Consider the quotient of a topological group $G$ by the relation "lies in the same connected component." You get a group $\pi_0(G)$ of connected components. This is equivalent to quotienting by the connected component of the identity (which incidentally proves that this subgroup is normal) but I think the equivalence relation is the more natural thing to think about, even if thinking about the normal subgroup is convenient.

One type of equivalence relation one can define on group elements is a double coset.

Another type of equivalence relation you see in group theory has to do with pairs of subgroups, rather than elements. If $1\leq M\trianglelefteq H \leq G$, then $(H,M)$ is referred to as a pair if $H/M$ is cyclic. $(H,M)$ is called "good" if $[g,H\cap g H g^{-1}]\not\subseteq M$ for any $g\in G\setminus H$. Lastly, if $(K,L)$ is another good pair, we call $(H,M)$ and $(K,L)$ related in G if there is a $k\in G$ so that $k^{-1}Hk\cap L=K\cap k^{-1}Mk$. (There are character theoretic definitions for all of these, too.) As it turns out, being "related" is an equivalence relation on the set of good pairs in G.

I don't know whether the second example is what you were looking for exactly when you talk about taking a quotient, but the interactions between these equivalency classes can be very interesting, and ends up motivating the study of a complicated class of groups called $M$-groups.