Note: Everything is commutative and unital
I am asked to determine the pushout in the category of $\mathbb{Z}$-algebras. So far, I have shown that the pushout is a tensor product modded by the relation $(f(n) - g(n))$. Is there something I overlooked?
This is basically correct, though your notation "$f(n)-g(n)$" is not very precise and I'm not sure you have the correct thing in mind. To be more precise, if $f:A\to B$ and $g:A\to C$ are homomorphisms of $\mathbb{Z}$-algebras, then their pushout is the quotient of the tensor product $B\otimes_\mathbb{Z} C$ by the ideal generated by all elements of the form $f(a)\otimes 1-1\otimes g(a)\in B\otimes_\mathbb{Z} C$ for $a\in A$. This quotient is also known as the $B\otimes_A C$, the tensor product of $B$ and $C$ as $A$-algebras.