I have a group $G=\mathbb{Z}_{36}$ with normal subgroups $K$ and $H$, where $H$ is also a normal subgroup of $K$. What I'm really supposed to find is a bijection between $G/K$ and $(G/H)/(K/H)$ by using the third isomorphism theorem. I have already calculated $G/K$, $G/H$ and $K/H$, but I do not really know how to calculate $(G/H)/(K/H)$.
Say that one of the elements in $G/H$ is $\{0,18\}$ and one of the elements in $K/H$ is $\{0,18\}$. Will one of the elements in $(G/H)/(K/H)$ be as follows? $$\{0,18\}+\{0,18\} = \{0+0,18+18\}=\{0,36\}=\{0\}$$
Another way I'm thinking it's possible to caluclate is $$\{0,18\}+\{0,18\} = \{0+0,0+18,18+0,18+18\}=\{0,18,18,36\}=\{0,18\}$$
Is any of these a correct way of calculating an element in $(G/H)/(K/H)$?
Though we really don't want to work with the explicit elements-as-cosets (it's too unwieldly), perhaps you want to really get a feel for them? Note that the Third Isomorphism Theorem gives you, explicitly (in conjunction with the first) the bijection you want. Trying to find one from scratch is an exercise in ignoring what you've already managed to prove.
So, first let's discuss how you want to use the Third Isomorphism Theorem to get this done. I'm using additive notation, since you are working with an additive group.
The elements of $G/H$ look like sets of the form $g+H = \{g+h \mid h\in H\}$. The set $g+H$ is the same as the set $g'+H$ if and only if $g-g'\in H$. The elements of $K/H$ are the sets where $g\in K$; that is, of the form $k+H$, with $k\in K$.
We have a map from $G/H$ to $G/K$, obtained by sending the set $g+H$ to the set $g+K$. Note that this is well defined because if $g+H=g'+H$, then $g-g'\in H\subseteq K$, so $g-g'\in K$, and therefore $g+K=g'+K$.
This map is a group homomorphism. The kernel consists of precisely the sets $g+H$ for which $g+K=0+K$. That is, the sets $g+H$ with $g\in K$. That is, $K/H$.
The First Isomorphism Theorem then tells you that you have a bijection $$\frac{G/H}{K/H} \to \frac{G}{K},$$ given by sending the element $(g+H) + (K/H)$ to the element $g+K$. That's your bijection.
Now, explicitly what are you doing, is that you have $G=\mathbb{Z}/36\mathbb{Z}$ (note: this is already a quotient! but you probably don't think of it as a quotient, but rather as consisting of number $0$ through $35$ with addition modulo $36$); I will denote the elements of $G$ by $\overline{k}$, with $0\leq k\lt 36$. And your $H$ is clearly $\{\overline{0},\overline{18}\}$. You don't tell us who is $K$, but let's say that it is $K=\{\overline{0},\overline{6},\overline{12},\overline{18},\overline{24},\overline{30}\}$.
The elements of $G/H$ are: $$\begin{array}{rcccl} \overline{0}+H &=& \{\overline{0},\overline{18}\} &=&\overline{18}+H,\\ \overline{1}+H &=& \{\overline{1},\overline{19}\} &=&\overline{19}+H,\\ \overline{2}+H &=& \{\overline{2},\overline{20}\} &=& \overline{20}+H,\\ &\vdots & &\vdots &\\ \overline{17}+H &=& \{\overline{17},\overline{35}\} &=&\overline{35}+H. \end{array}$$ And we have $$K/H = \{ \overline{0}+H,\overline{6}+H,\overline{12}+H\}.$$ (Because $\overline{18}+H=\overline{0}+H$, $\overline{24}+H=\overline{6}+H$, and $\overline{30}+H = \overline{12}+H$.)
Since the elements of $(G/H)/(K/H)$ are of the form $(\overline{a}+H)+(K/H)$, we have: $$\begin{array}{rcccccl} (\overline{0}+H)+(K/H) &=& \{ \overline{0}+H,\overline{6}+H,\overline{12}+H\} &=& (\overline{6}+H)+(K/H) &=& (\overline{12}+H)+(K/H)\\ (\overline{1}+H) + (K/H) &=& \{ \overline{1}+H, \overline{7}+H, \overline{13}+H\} &=& (\overline{7}+H) + (K/H) &=& (\overline{13}+H)+(K/H)\\ (\overline{2}+H) + (K/H) &=& \{ \overline{2}+H, \overline{8}+H, \overline{14}+H\} &=& (\overline{8}+H) + (K/H) &=& (\overline{14}+H)+(K/H)\\ (\overline{3}+H) + (K/H) &=& \{ \overline{3}+H, \overline{9}+H, \overline{15}+H\} &=& (\overline{9}+H) + (K/H) &=& (\overline{15}+H)+(K/H)\\ (\overline{4}+H) + (K/H) &=& \{ \overline{4}+H, \overline{10}+H, \overline{16}+H\} &=& (\overline{10}+H) + (K/H) &=& (\overline{16}+H)+(K/H)\\ (\overline{5}+H) + (K/H) &=& \{ \overline{5}+H, \overline{11}+H, \overline{17}+H\} &=& (\overline{11}+H) + (K/H) &=& (\overline{17}+H)+(K/H) \end{array}$$
Now, if we really wanted to expand that, we would need to express each of the elements in the coset as sets. So for example, $$\begin{align*} (\overline{1}+H)+(K/H) &= \{ \overline{1}+H, \overline{7}+H, \overline{13}+H\}\\ &= \Bigl\{ \{\overline{1},\overline{19}\} , \{\overline{7},\overline{25}\}, \{\overline{13},\overline{19}\} \Bigr\}, \end{align*}$$ a set of sets. (And if we were looking at $G$ as a quotient, it would be a set of sets of sets of integers).
Which is pretty much why you don't want an explicit description. You want to work with the cosets symbolically, not directly.