Suppose p is a prime. Find all the zero divisors in $\displaystyle Z_p$×$\displaystyle Z_{p^2}$
Any member of $\displaystyle Z_p\times Z_{p^2}$ is of the form (a, b) where a is in $\displaystyle Z_p$, b is in $\displaystyle Z_{p^2}$ and multiplication is coordinate wise. That is, to be a zero divisor we must have $\displaystyle (a, b)(c, d)= (ac, bd)= (np, mp^2)$ for n and m integers. Since p is prime, at least one of a or c is a multiple of p and so is congruent to 0 modulo p. Either one of b or d is a multiple of $\displaystyle p^2$, and so congruent to 0 modulo $\displaystyle p^2$ or both b and d are multiples of p.
I don't know what you mean by "the first set" and "the second set". There is only one set to be found here: the zero divisors of $\displaystyle Z_p\times Z_{p^2}$. Yes, since p is a prime, $\displaystyle Z_p$ has no zero-divisors. So any zero-divisor of $\displaystyle Z_p\times Z_{p^2}$ must be of the form (0, a) where a is a zero-divisor of $\displaystyle Z_{p^2}$. What are the zero-divisors of that?