How do I show that $T = {\bf Th}({\bf Z}, +)$ has uncountably many pairwise nonisomorphic countable models.
I know how to shew that ${\rm Th}({\bf Z}, +, \cdot)$ has uncountably many pairwise nonisomorphic countable models; to every infinite set of standard primes there is a model with an element which is divisible by exactly those primes, and each model corresponds to countably many such sets.
The only thing you're missing is that for a standard prime $p$, the property "$p$ divides $x$" is definable in the language with just $+$: $$\exists y (y+y+\dots+ y= x)$$ where the sum is of $p$ copies of $y$.