I'm having trouble understanding how to locate the zeros of a polynomial using the beginning of the argument.
I need to locate the zeros of this polynomial $p(z) = z^9 - 8z^2 + 5$, but I don't know how to calculate the change of the argument of the function.
I am not sure what you mean by "using the beginning of the argument", but something like the reasoning below may help.
E.g. let $r\ge1$ be any real number such that $r^9>8r+5$ and let $\gamma:[0,2\pi]\to\mathbb C$ be the path $\gamma(t)=re^{it}$. Then $$|\gamma(t)-p\circ\gamma(t)| =|8\gamma(t)^2-5| \le8|\gamma(t)|^2+5 =8r^2+5 <r^9 =|\gamma(t)-0|.$$ Therefore the winding number $n(p\circ\gamma,0)$ is equal to $n(\gamma,0)=9$. This means all zeroes of $p$ lie inside the open disc $|z|<r$.
The example above actually does not justify the use of the argument principle. It is easy to reach the same conclusion without it: if the $r$ we pick is $\ge1$, then $R^9>8R+5$ for every $R\ge r$. Therefore all roots of $p(z)=0$ must lie inside $|z|<r$, otherwise if we put $R=|z|$ we will have $R\ge r$ and hence $R^9=|z|^9=|8z-5|\le8|z|+5=8R+5<R^9$, which is a contradiction.
However, if we pick a more sophisticated path $\gamma$ in the earlier example, it is possible to locate the zeroes of $p$ more precisely using the argument principle.