Determine the generator for the cyclic group formed by the solutions of $x^9 = 1$. 
Find the solutions to $x^9 = 1$ and determine the generator for the cyclic group formed by the solutions.

The equation can be factored as $(x^3 - 1)(x^6 + x^3 + 1) = 0$ and the solutions are $$\begin{align*}
    x &= 1\\
    x &= -\sqrt[\leftroot{1}\uproot{3}9]{-1} \\
    x &= (-1)^{2/9} \\
    x &= -\sqrt[\leftroot{1}\uproot{3}3]{-1} \\
    x &= (-1)^{4/9} \\
    x &= -(-1)^{5/9} \\
    x &= (-1)^{2/3} \\
    x &= -(-1)^{7/9} \\
    x &= (-1)^{8/9}
\end{align*}$$ but I'm unsure how to determine the generator. How are these found generally and what would be my options to start determining it?

Your elements are basically $(e^{ik\pi/9})_{k=0}^8$. They form the cyclic group of $9$ elements under ordinary multiplication. The most obvious generator is $r=e^{i\pi/9}$ (with $k=1$).
However, any such element with $k$ relatively prime to $9$ (e.g. $4,5,8$) would generate the group as well. For example, starting with $r^4$, you get a cycle
$$
\left< r^4,r^8,r^{12} = r^3,r^7,r^{11}=r^2,r^6,r^{10} = r,r^5, r^9 = 1 \right>
$$
