What is the standard notation to represent the set of primes?

First: $\mathbb{Z}_p$ does not refer to the set of primes. Depending on the context, it either refers to $\mathbb{Z}/p\mathbb{Z}$ or the $p$-adic integers.

The most common notation that I have seen for the set of primes is $\mathbb{P}$; however, it is not universally used, and so you should make sure to define it whenever you use it.

$\mathbb{P}$ seems to be fairly common.

It is surprisingly common to eschew any such notation and, instead, rely on a convention that $p$ (and $p'$, $p_n$ etc) always denotes a prime number.

In fact, you can see that convention used in a recent question right here.