John Hempel, in his book $3$-manifolds, shows that if $G$ is a finitely generated abelian group which is a subgroup of the fundamental group of a closed $3$-manifold, then $G$ is one of $\mathbb Z$, $\mathbb Z\oplus\mathbb Z$, $\mathbb Z\oplus\mathbb Z \oplus\mathbb Z$, $\mathbb Z_p$ or $\mathbb Z\oplus\mathbb Z_2$. This is theorem 9.13 in the book.
He also proves, in theorem 9.14, that an abelian group which is not finitely generated and a subgroup of the fundamental group of a $3$-manifold, then it is isomorphic to a subgroup of $\mathbb Q$ (and proposes, as an exercise, to show that all such groups in fact occur)