For the first part I think I have it:
Let $(m \times n) \in (M \times N)$ and $(g\times h) \in (G \times H)$
Then $(g \times h)(m \times n)(g \times h)^{-1} = (g + m - g \times h + m - h) = (m \times n) \in (M \times N)$.
So $(g \times h)(m \times n)(g \times h)^{-1}$ is a subset of $(M \times N)$ which implies $(M \times N)$ ⊴ $(G \times H)$.
For the second part I was planning on using the first homomorphism theorem. My issue is I'm not sure what the identity element is in (G/M) × (H/N). I would think it is M x N but I'm not even sure what the group operation is for that group.
Any help would be appreciated.
Thanks