I'm trying to get the 'big picture' of when each of these might come in handy.
So far, it seems that we would prefer to be able to use the Fundamental Theorem of Finitely Generated Abelian Groups (FtFAG) whenever possible since this provides more information on how to break up the structure of $G$ into simpler groups that we can handle. This is obviously only available to us when $G$ is finitely generated.
If, however, $G$ is not finitely generated, we could turn to using Free Abelian Groups in order to understand something (likely less informative compared to what FtFAG would have been able to provide) about $G$'s structure. This, for example, would be made possible by defining a homomorphism $h$ between a free abelian group $F$ with infinite basis (of same cardinality as some generating set for $G$?) and then investigating the group $F/ker(h)$ (after applying the first isomorphism theorem, which is possible since $h$ is surjective). The idea is that $F$, $ker(h)$ provide more information to work with (two groups) compared to $G$ (one group), thus providing some form of 'breaking up' $G$ into simpler groups, as in FtFAG.
Would this intuition be correct? (if I had to sum it up -- would prefer to use FtFAG, but if cannot, turn to use free abelian groups?).
Is there anything else worth noting along these lines?