Note: My question is a lot more general. Can you provide me with an answer about what "classify" implies when it comes to groups and what are the steps that one should follow to carry through with this process?
Attempt:
Let $G=\mathbb{Z}_{108}$
According to the Fundamental Theorem of Finite Abelian groups, if we write $G$'s order as $$ 108=2^23^3 $$ we can infer that $G$ is isomorphic to one of the following direct products of cyclic groups:
\begin{align*} 1.&\quad \mathbb{Z}_{4} \times \mathbb{Z}_{27} \\ 2.&\quad \mathbb{Z}_4 \times \mathbb{Z}_3 \times\mathbb{Z}_9 \\ 3.&\quad \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_{27} \\ 4.&\quad \mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \\ 5.&\quad \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times\mathbb{Z}_9 \\ 6.&\quad \mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \\ \end{align*} Does this process go any further or after this observation $G$ is considered as classified?
What you have done is correct. However, you are not yet finished! You still need to show which of these direct products actually represents $G$, since those products aren't all isomorhpic to each other. Consider the order of each element of $G$. If you find one with, say, order 27, then $G$ is isomorphic to either $1.$ or $3.$ in your list. Next, you can try to find an element with order 2. If you find one, you are done, and the result is $3.$ . If you instead find an element with order 4, then $G$ is isomorphic to $1.$ .