Does a simple symmetric random walk on $\mathbb{Z^{2}}$ reach each vertex?

As you say, the standard random walk on $\mathbb{Z}^2$ is recurrent. There are I think several proofs of them, none of which are very easy. You might want to look at this old thread on StackExchange:

Intuitive reason why a simple symmetric random walk is recurrent on $\Bbb Z^2$ and transient on $\Bbb Z^3$.

Once you know that a random walk on $\mathbb{Z}^2$ is recurrent, you know that any random walk will almost certainly return to the origin (the starting point) infinitely often. Every time it returns, it goes in one of four directions. Almost certainly, it will go into every one of these directions infinitely often. Hence the points around the origin are reached infinitely often. Now inductively work your way outward from the origin.