I am trying to show that if $f, g: (X,x) \to (Y,y)$ such that $f \cong g$, then the induced homomorphisms $f_{*}, g_{*}: \pi_1(X,x) \to \pi_1(Y,y)$ are equal.
I proved that for any $h \in \pi_1(X,x)$, we have $f_{*}(h) = [f \cdot h] = [g \cdot h] = g_{*}(h)$ since $f \cdot h \cong g \cdot h$. Is this sufficient or am I forgetting to prove something else?
Yes, this is basically correct. To be pedantic, you should really write $[h]$ instead of $h$ for your element of $\pi_1(X,x)$. Note also that it is important that all homotopies you talk about are basepoint-preserving, since you need basepoint-preserving homotopies (of maps from $S^1$, or homotopies preserving both endpoints if you consider them as maps from $[0,1]$) to get equality in $\pi_1$.