Describing explicitly the MyHill-Nerode classes created by a language
Your classes are correct, and one can also describe them in words: words containing no $a$, words containing a single $a$, words containing at least two $a$s. When described in this fashion, it is clear that the every word belongs to exactly one class.

However, there is no reason to use the Myhill–Nerode criterion to prove that a language is regular. Instead, you can use its vast generalization and give a DFA or NFA for the language. In the other direction, there is no need to describe all classes, only infinitely many; this already shows the language is not regular.