Find generator of multiplicative group of $\mathbb{F}_{27}$

Start with finding the order of, say, $t$. It can't be $2$ (since $t^2-1\neq0$), so it's either $13$ or $26$. So either $t$ or $t^2$ is the one you're looking for.

Nothing wrong with Arthur's answer. But the Frobenius automorphism (or Freshman's dream) makes the calculation very easy:

Let $t$ be a zero of $x^3-x-1$. Then $t^3=t+1$. Consequently $$t^9=(t^3)^3=(t+1)^3=t^3+1=t+2=t-1.$$ Implying that $$t^{13}=t^{1+3+9}=t(t+1)(t-1)=t^3-t=1.$$ As $13$ is a prime and $t\neq1$, we can conclude that $t$ has order $13$.

It may be worth noting that $-t$ is then of order $26$, just in case you need a generator of $\Bbb{F}_{27}^*$.