Let $G$ and $H$ be groups. Let $N_1$ be a normal subgroup of $G$ and let $N_2$ be a normal subgroup of $H$. Is the following statement true? $$(G \times H)/(N_1 \times N_2)= G/N_1 \times H/N_2$$
More specifically, I wonder if $(\mathbb{Z} \times \mathbb{Z})/(3\mathbb{Z} \times \mathbb{Z})= \mathbb{Z}/3 \mathbb{Z}$, but I'm also interested in the general statement.
If by $=$ you mean "equal", the answer is "NO". If you mean "isomorphic", the answer is "YES"....provided that you meant $H/N_2$. Just apply the first isomorphism theorem to the group morphism $$(g,h)\in G\times H\mapsto (gN_1,hN_2)\in G/N_1\times H/N_2.$$