Considering
$$0\to \Bbb Z \to \Bbb Z\to \Bbb Z/(2)\to 0$$ We want to show that the tensor product is not left exact by tensoring with $\Bbb Z/(2)$, which apparently gives us what we want.
But I did this, and I get
$$0\to \Bbb Z/(2)\overset{\sim}{\to}\Bbb Z/(2)\overset{0}{\to} \Bbb Z/(2)$$
Where I can't see why this isn't exact? Is there a reason why I can't have the zero map on the right?
The question title is "tensoring is not left exact," so you should probably be looking for failures in exactness towards the left of the second sequence.
The map $\Bbb{Z}\rightarrow\Bbb{Z}$ in the original sequence is multiplication by $2$. You need to figure out what the induced map is after tensoring by $\Bbb{Z}/2\Bbb{Z}$. (The important thing is that it is NOT an isomorphism, as you initially thought.)