Help on notation: $\mathbb{Z}/n\mathbb{Z}$ vs. $\mathbb{Z}_n$

It depends on the textbook/paper author, but often $\mathbf{Z}/n\mathbf{Z}$ and $\mathbf{Z}_n$ mean the same thing.

A word of caution, however: using the notation $\mathbf{Z}_n$ to mean $\mathbf{Z}/n\mathbf{Z}$ can cause confusion, because $\mathbf{Z}_p$ is also used to denote the p-adic integers. Thus, many mathematicians (especially number theorists) reserve the shorter notation for p-adics and use the long notation for the finite cyclic groups.

Edit: Just now saw your second question. The answer is that, indeed, $\mathbf{Z}/p\mathbf{Z} = GF(p)$, where $p$ is prime.

If $n$ is a prime number, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are isomorphic (in fact I would simply define $GF(n)=\mathbb{Z}/n\mathbb{Z}$ when $n$ is a prime number).

However, if $n$ is some power of a prime number, say $n=p^k$ for $k\geq 2$, then $\mathbb{Z}/n\mathbb{Z}$ and $GF(n)$ are not the same.

The notations are equivalent if the author has been careful enough to tell you that by $Z_n$ she means "the integers modulo $n$." If she has not been careful than you have to study the context to decide whether the author means the integers modulo $n$ or something else.

By the way, $Z/nZ$ is not just a quotient group, it's a quotient $\it ring$ (if you haven't studied rings and ideals yet, you have something to look forward to!).