Basically you do what Davide Giraudo suggested. We use the definition as given on the EOM
An upper semicontinuous function $v:\Omega \to\mathbb{R}\cup \{-\infty\}$ is called subharmonic if for every $x_0 \in \Omega$ and for every $r > 0$ such that $\overline{B_r(x_0)} \subset \Omega$, that $ v(x_0) \leq \frac{1}{|\partial B_r(x_0)|} \int_{\partial B_r(x_0)} v(y) \mathrm{d}y $
Now, using that $f$ is convex, Jensen's inequality takes the following form:
$ f\left( \frac{1}{|\partial B_r(x_0)|} \int_{\partial B_r(x_0)} v(y)\mathrm{d}y \right) \leq \frac{1}{|\partial B_r(x_0)|}\int_{\partial B_r(x_0)} f\circ v(y) \mathrm{d}y $
If $u$ is harmonic, we have that
$ \frac{1}{|\partial B_r(x_0)|} \int_{\partial B_r(x_0)} u(y)\mathrm{d}y = u(x_0) $
so combining the two facts you have that
$ f\circ u(x_0) \leq \frac{1}{|\partial B_r(x_0)|}\int_{\partial B_r(x_0)} f\circ u(y) \mathrm{d}y $
which is precisely the definition for $f\circ u$ to be subharmonic.