Suppose that the number of homomorphisms from $G$ to $H$ is n. How many homomorphisms are there from $G$ to $H \oplus H$? When H is abelian, how many homomorphisms are there from $G \oplus G$ to $H$?
If we let $g_1,g_2 \in G$ We have $\phi_1(g_1g_2)=\phi_1(g_1)\phi_1(g_2)$ ... $\phi_n(g_1g_2)=\phi_n(g_1)\phi_n(g_2)$.
Now in $H \oplus H$ we have that $\phi_1(g_1)$ is $(h_1,h_2)$. Any hints on how to approach this?
Given any two $\phi,\psi:G\to H$ homomorphisms, you get a homomorphism $G\to H\oplus H$ given by $g\longmapsto (\phi(g),\psi(g))$. Conversely, given $\beta:G\to H\oplus H$ homomorphism, since the projections $\pi_j:H\oplus H$ given by $\pi_1(x,y)=x$, $\pi_2(x,y)=y$, you get two homomorphisms $\pi_j\circ\beta:G\to H$ that reverse the construction from the beginning.
This shows that $\operatorname{Hom}(G,H\oplus H)=\operatorname{Hom}(G,H)\times\operatorname{Hom}(G,H)$.