Cardinality of the set of all pairs of integers

Natural Numbers: There are many pairing functions that map $\mathbb{N}\times \mathbb{N}$ bijectively to $\mathbb{N}$. A simple example is the mapping $f$ such that $f(a,b)=2^{a-1}(2b-1)$. For every positive integer $y$ can be uniquely expressed as a power of $2$ times an odd integer.

Integers: If you want a mapping $g(x,y)$ that maps $\mathbb{Z}\times \mathbb{Z}$ bijectively to $\mathbb{N}$, it is simplest to split the work into two parts.

Let $\phi$ be any mapping that maps $\mathbb{Z}$ bijectively to $\mathbb{N}$. For a concrete example of such a mapping, let $\phi(t)=2t+2$ if $t \ge 0$, and let $\phi(t)=-(2t+1)$ if $t \lt 0$. The non-negative integers are sent to the even integers $\ge 2$, and the negative integers are sent to the positive odd integers.

Then the mapping $g(x,y)=f(\phi(x),\phi(y))$ works, where $f$ is any bijective map from $\mathbb{N}\times \mathbb{N}$ to $\mathbb{N}$. For example, we can use the mapping $f$ of the first paragraph, or the Cantor pairing function.

Remark: For most purposes, there is no particular virtue in having an explicit bijection, as long as we can prove that a bijection exists.

You simply haven’t yet found a function that works. One that does is the Cantor pairing function, which is described quite well in the Wikipedia article to which I linked.