How many functions are there from $\mathbb Z$ to $\mathbb Z$? Repeating the question,

How many functions are there from $\mathbb Z$ to $\mathbb Z$?

A function $f \colon A \to B$ is a subset of $A \times B$ satisfying
$$(a,b) = (a,c) \qquad \Rightarrow \qquad b = c,$$
so it's enough (maybe) to look at subsets of $\mathbb Z \times \mathbb Z$. We know $|\mathbb Z \times \mathbb Z| = |\mathbb Z|$, and that the number of subsets of $\mathbb Z$ is $2^{|\mathbb Z|}$, but this counts finite subsets as well. Unsure of where to proceed from here.

You’ve made a start. As you say, each function from $\Bbb Z$ to $\Bbb Z$ is a subset of $\Bbb Z\times\Bbb Z$, so there are at most $2^{|\Bbb Z|}$ of them. To finish the argument you could prove that there are also at least $2^{|\Bbb Z|}$ such functions by actually finding that many that you can clearly identify as distinct.
HINT: For each $A\subseteq\Bbb Z$ let
$$f_A:\Bbb Z\to\Bbb Z:n\mapsto\begin{cases}
1,&\text{if }n\in A\\
0,&\text{otherwise}\;.
\end{cases}$$
