Let $Z^i_n=(Z^i_n(1), \cdots, Z^i_n(d))$, $ n \geq 0$, be a $d$-type supercritical branching process in an independent and identically distributed random environment $\xi=(\xi_0, \xi_1,\cdots )$, starting with one initial particle of type $i \in \{1, \cdots, d\}$. We study asymptotic properties of $Z_n^i (j)$, the $j$-type population size of generation $n$, for each $j\in \{1, \cdots, d\}$, as $n$ goes to infinity. For $Z_n^i(j)$ we establish a Berry-Esseen type bound for the rate of convergence in the central limit theorem and a moderate deviation principle. As an important ingredient of the proofs, we also demonstrate the existence of the harmonic moments (which is of independent interest) of the normalized population size $Z_n^i(j) / \mathbb E_\xi Z_n^i(j)$, uniformly in $n \geq 1$, where $\mathbb E_\xi $ stands for the conditional expectation given the environment $\xi$.