Can someone please verify whether my proof is okay? It looks very tedious and possibly stating unnecessary statements. I might also be missing some important facts.
Prove that $p$ and $q$ are logically equivalent if and only if $p\leftrightarrow q$ is a tautology.
Assume $p$ is logically equivalent to $q$. Then every truth value gives $p$ and $q$ the same truth values. Then $p\leftrightarrow q$ is a tautology.
Assume $p\leftrightarrow q$ is a tautology. Then $p$ and $q$ have the same truth values. Then it must be that $p$ is logically equivalent to $q$.
"Assume p is logically equivalent to q. Then every truth value gives p and q the same truth values."
That sounds fine. 'p' and 'q' have the same truth value by definition.
"Then p↔q is a tautology."
You might do better to refer to the values of ($\bot$↔$\bot$) and ($\top$↔$\top$) specifically. And you might do well to note that no other truth values than $\top$ and $\bot$ exist for how ↔ gets understood, because there exist three-valued logical systems where there exist connectives for a similar concept, but the same truth values for both 'p' and 'q' do not guarantee that (p↔q) is a tautology.
The same goes for the second part.