Give a reason why $\mathbb Z_{64}\times \mathbb Z_4$ and $\mathbb Z_{64}\times \mathbb Z_{2}\times \mathbb Z_{2}$ are not isomorphic.

That was my first thought of a way to do it.

Depending on where you are in your learning you might be expected to show that if the first component is other than $0$ or $32$ the order is greater than $2$ (and similarly for the second component in the $\mathbb Z_4$ case). But the reasoning is completely sound.

You might also identify a subgroup $\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$ in one case rather than the other. Orders of elements and subgroups are two potentially distinguishing features to look out for.

Draw the lattice of subgroups, maybe. The lattice of the Klein four group is not a total order, but that of the cyclic group of order four is. Then note how doing $G\times -$ changes the lattice.