How do I find a partition of an equivalence relation? Say I have the function:
$$x\,R\,y \iff y = 3^k$$ for some $k \in \mathbb Z$ and the set is:

$$A = \{1,1/3,1/27,1/4,3,1/36 , 2,2/9,9/4, 5\}$$

So in this scenario, how do I find the partitions of the set?

Knowing a fairly example should allow me to know how to do it in the future.
An equivalence relation is always partition. In general you want to put it in the form of a set. Here, you can say,

$$A_{r}= \{(x,y) | y = \frac{x}{3^k}\}$$

Where $A_r$ denotes the equivalent partition corresponding to your relation. Also, $k$ is assumed to be an integer.
A partition of a set $A$ is a collection of $a_i \subset A$ such that each  $a_i$ is disjoint from the others and $\bigcup a_i = A$.

An equivalence relation always partitions a set. Given a set $A$ and some relation $R$, the set of equivalence classes $A/R$ is a partition of $A$. Each member of $A/R$ is a subset of $A$ such that each member is $R$ related.

Since your given set is a finite set, perhaps the simplest approach would be to start with an element and determine all the other elements that are $R$ related to it. The set of all these elements is one equivalence class in the partition. Then pick another element from $A$ that is not in this first equivalence class, and form another class. Once all the elements of $A$ have been grouped, you will have formed the partition of $A$ that you are looking for.