There is no simple classification theorem for countably infinite Abelian groups.
There are many different theorems in the literature that make the sentence above precise in various ways, but here's one that I think is pretty compelling: Hjorth proved that the isomorphism relation for countable abelian groups (even torsion-free ones) is non-Borel.
Another result along the same line is a result of Thomas, which says that if you look at torsion-free countable abelian groups of finite rank, there is no Borel map that takes invariants that work for rank $n$ groups and produces invariants that work for rank $n+1$ groups. Thus the complexity problems gets strictly harder as the rank increases.
The results I've mentioned here show that the classification problem is hard even for torsion-free countable abelian groups, so you can imagine the situation certainly isn't any better for arbitrary ones.