To find the ML, am I right in thinking you take the log of the function then differentiate it with respect to p?
I am a bit confused as to what to do with the data given and am not sure how to find f(0) and f(1). Can someone explain how to go about answering questions like this?
The likelihood is
$$L(\{ z_n \} \mid p)=\prod_{i=1}^n f_Z(z_i \mid p).$$
Given data $\{ z_n \}$, the MLE $\hat{p}$ is the maximizer of $L$ in the variable $p$. It is usually easier to compute the maximizer of $\log(L)$, which is the same because $\log$ is strictly increasing. That last maximum can be computed by differentiation, so you're looking to solve the equation
$$\sum_{i=1}^n \frac{\frac{\partial f_Z}{\partial p}(z_i \mid p)}{f_Z(z_i \mid p)}=0.$$