How do I show that $\mathbb{Z}$ is not isomorphic to $\mathbb{Z} \times \mathbb{Z}_3$? More generally, what is the best tactic for proving that two groups are not isomorphic to each other when they are of the same cardinality?

In this case, it's easy: $\mathbb{Z}\times\mathbb{Z}_3$ has an element whose order is $3$, whereas $\mathbb Z$ has no such element.

In general, thinking about the orders of the elements is a good approach.