What is a simple example of a free group?

The simplest example is $(\mathbb{Z},+)$ with the basis $X=\{1\}$. Let $\varphi$ be any function from $X$ into a group $G$ and let $g=\varphi(1)$. Now, define $\tilde{\varphi}(m)=g^m$, for each integer $m$. Then $\tilde\varphi$ is a group homomorphism. Furthermore, it is the only group homomorphism from $(\mathbb{Z},+)$ into $G$ such that $\tilde\varphi(1)=\varphi(1)$.

The simplest non-abelian example is $F_2$, with generating set $S=\{a,b\}$. The elements are just reduced words of the form $a^{r_1}b^{s_1}\cdots a^{r_n}b^{s_n}$. In fact, $F_2$ can be shown to be isomorphic to a matrix subgroup of $SL_2(\mathbb{Z})$, namely the subgroup generated by $$ a = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \ \ \ \ \ b = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}, $$ see the Ping-Pong Lemma.

You have a set $X$. The elements of $F_X$ the free group on $X$ are "reduced words" in $X$; finite sequences of symbols $x$ and $x^{-1}$ for $x\in X$ where no $x$ and $x^{-1}$ are adjacent and including the empty word, which I'll write as $1$ as it is the identity. So for $X=\{x,y,z\}$ reduced words include $1$, $x$, $y$, $z^{-1}$, $yx$, $xzx^{-1}$, $xxyx^{-1}z^{-1}xzy^{-1}z$ etc. To multiply, concatenate, and cancel any $xx^{-1}$s or $x^{-1}x$s.

If you have $X=\{x\}$ with one element, the reduced words are $1$, $xx\cdots x$ (with $n$ $x$s) and $x^{-1}x^{-1}\cdots x^{-1}$ (with $n$ $x^{-1}$s). If we write these instead as $x^0$, $x^n$ and $x^{-n}$ we see that $F_X\cong\Bbb Z$.

Rather perversely, in your notation you have chosen $0$ to be the element. This results in reduced words like $0000$ and $0^{-1}0^{-1}0^{-1}$. These can cause confusion...