Finding the equivalence classes of a language I'm doing a problem where I need to find the $≡_A$ equivalence classes of the language
$$A = \{ 0^{n}x \mid n \in \mathbb Z^+, x \in \{0, 1\}^*, \text{ and } \#_0(x) ≥ n \}. $$

The best way I've learned to find the equivalence classes from a formal language is to create an automation and minimize it. Is there a conventional way of finding the equivalence classes of a language more quickly and intuitively? Because I do not know the DFA for this problem. Thanks for the help!
Your language is $01^*0(0+1)^*$, i.e., all words starting with 0 and containing at least one more 0.

Indeed, a word of the form $0y0z$ is in $A$ since we can take $n=1$ and $x = y0z$. Conversely, if $w = 0^nx \in A$ then $w$ contains at least $2n \geq 2$ many 0's, and starts with a 0.

Therefore the Myhill–Nerode equivalence classes of your language are:
$$
\epsilon, 1(0+1)^*, 01^*, 01^*0(0+1)^*.
$$