I'm just learning about finite, countably infinite, and uncountable sets. My question is, which of the three categories would this fall into:
{3^n|n ϵ Z}
I first thought that it would uncountable, but I don't think it's cardinal number would be larger than that of the set of all natural numbers. So I'm thinking most likely that it is countably infinite but am not too sure.
Your intuition is correct, the set is countably infinite.
There is even a fairly simple bijection $\varphi$ from the set $\mathbb{N}$ of natural numbers to your set. The details depend slightly on whether we view the natural numbers as including $0$ or not. We will consider $\mathbb{N}$ as including $0$. A small modification of $\varphi$ will take care of things if you view $\mathbb{N}$ as not including $0$.
Let $\varphi(0)=3^0$. For positive even $n$, let $\varphi(n)=3^{n/2}$. For positive odd $n$, let $\varphi(n)=3^{-(n+1)/2}$. Then $\varphi$ is a bijective map from $\mathbb{N}$ to the set $\{3^n|n\in \mathbb{Z}\}$.