Finding infinitely many Myhill–Nerode equivalence classes I need to prove the following languages are not regular and I'm having a
tough time identifying the equivalence class that can have an infinite
index.
How would I go about doing these? I know how to prove it after finding
the equivalence class but any tips on how to identify the one that has
the infinite index?
I know I should find the equivalence class where there are distinct
extensions for each string belonging to $L$, but could anyone help me
out on these 2:

$0^m  1^n$ where $m$ is even or $m$ is greater than $n$

$0^m  1^n$ where $\text{gcd}(m,n)=1$.
Let's consider your second example (the first example is easier). We will be interested in
$$
F(m) = \{ n : 0^m 1^n \in L \} = \{ n : \gcd(m,n) = 1 \}.
$$
If $F(m_1) \neq F(m_2)$, then $m_1$ and $m_2$ belong to two different equivalence classes.
From $F(m)$ you can recover the set of primes dividing $m$: $p \in F(m)$ iff $p \not\mid m$. In particular, this shows that if $p_1 \neq p_2$ are two different primes, then $F(p_1) \neq F(p_2)$. Since there are infinitely many primes, we conclude that there are infinitely many different equivalence classes.
The first example can be analyzed in a similar way (considering only odd $m$).