Maximal (closed) subgroups of semisimple Lie groups have been
first classified by E.B. Dynkin (see the reference above). There is indeed a way to "reduce"
the problem to the case of a simple Lie group, but it is more complicated than just
to take the direct product of the maximal subgroups of the simple factors.

In the example $G=SU(n) \times SU(n)$, or in the example $SU(n)\times_{\mathbb{Z}_n}
SU(n)=SU(n)\otimes SU(n)$, we would indeed start with the maximal subgroups of $SU(n)$, which are given by:

(a) $SO(n)$

(b) $Sp(m), m=2n$

(c) $S(U(k) \times U(n-k))=\lbrace (A,B)\in U(k)\times U(n-k) \mid \det(A)\det(B)=1\rbrace$ for $1\le k\le n-1$.

(d) $SU(p)\times_{\mathbb{Z}_d} SU(q)$, where $d=gcd(p,q)$, $pq=n$, $p,q\ge 3$

(e) $\rho(H)$, $H$ simple, $\rho \in Irr_{\mathbb{C}}$ of degree $n$.

Then we should proceed as in Theorem 5.9, 5.10 etc. of the following (modern) reference:

Fernando Antoneli, Michael Forger and Paola Gaviria,
*Maximal Subgroups of Compact Lie Groups*.

see http://arxiv.org/pdf/math/0605784v3.pdf

I am a bit too lazy to write out the result. Anyway, you don't have to read Dynkin (although this is interesting !).