Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes.
Proof: Let $S$ be a relation on S defined by $G \sim H$ iff $G ≈ H$. Show reflexive, symmetric, transitive. It is obvious that $G≈G$ so $G\sim G$. Assume $G\sim H$, thus $G≈H$, which implies that $H≈G$, so $H\sim G$, thus symmetric. Assume $G\sim H$ and $H\sim K$, for $K$ a group in $S$. Thus $G≈H$ and $H≈K$. (I have already shown that if $G≈H$ and $H≈K$, then $G≈K$). From earlier this implies that $G≈K$, so $\sim$ is an equivalence relation. Thus $S$ is partitioned into isomorphism classes. QED.
The proof doesn't do anything except reformulating. You really have to tell and convince the reader of your proof that your arguments are correct. For example, as for reflexivity, write down an isomorphism $G \to G$. As for symmetry, given an isomorphism $G \to H$, construct an isomorphism $H \to G$. If you have already proven transitivity, that's OK. By the way, there is no need to choose an extra symbol ~ for what is just ≈, just work with ≈ or the classical symbol $\cong$ for isomorphism.