What's the basic steps to show a set is denumerable? For example, $\mathbb{N}$ is denumerable.

A set $A$ is denumerable iff there exists (at least) one injective function $f:A\to\Bbb N$ and one injective function $g:\Bbb N\to A$. For $A=\Bbb N$ the identity is such a function (so is "add one" or "multiply by two", so there are many). For any other set you have to look for such functions.