While the literature has many variations, all the published proofs I know induct on the number of generators. Thus they start will an abelian group $A$ and build up a direct sum of cyclic groups within it. Part of the difficulty involves the occasional need to remove a direct summand and replace it with another in order to proceed.
My question: Does any published proof induct on the relations?
With no relations, one has a free abelian group, so a direct sum of infinite cyclic groups. Adding a relation induces a quotient, and the idea is to see how the direct sum decomposition behaves with respect to the quotient process.
To illustrate one important step, I'll explain how to argue that the torsion subgroup of a finitely generated abelian group has a free abelian complement. Assume this by induction and add a new relation. The relation has the form $f+t$, where $f$ belongs to the free abelian summand and $t$ to the torsion subgroup. Say $t$ has order $m$. Then $m(f+t)=mf$. If $f$ does not equal $0$, choose a basis for the free group that makes $mf$ a multiple of a basis element (standard stuff). Now modding out by $mf$ produces torsion and reduces the rank. Then modding out by $f$ takes a quotient of the torsion.
Analyzing the torsion part turns out only just a little more invovled.