I have gotten to the point where I found the joint pdf of $Z_1$ and $Z_2$ is
$h_{1,2}(z_1,z_2) = 2f(z_1)f(z_1+z_2)$,
but I do not know where to go from here because when I assume the independence of $Z_1$ and $Z_2$, I cannot get anywhere:
$h_{1,2}(z_1,z_2) = 2f(z_1)f(z_1+z_2) = h_1(z_1)h_2(z_2)$,
where $h_1(z_1)$ and $h_2(z_2)$ represent the marginal pdf of $Z_1$ and $Z_2$, respectively.