How is $\mathbb{F}_4$ generated? I know $\mathbb{F}_4$ is a field while $\mathbb{Z}/(4)$ is just a ring. So how is $\mathbb{F}_4$ generated? 
Complement: So what are the elements like in $\mathbb{F}_4$ like? Are they $\{0,1,x,x+1\}$? Is every field $\mathbb{F}_k$ has $k$ elements no matter whether $k$ is a prime?

The answer of @MarceloBielsa is perfect, but I like an approach like what OP was working towards: the elements of $\mathbb F_4$ are $\{0,1,\omega,\omega+1\}$, where $\omega^2=\omega+1$, $(\omega+1)^2=\omega$, and $\omega(\omega+1)=1$. I think that that’s enough for filling in the multiplication table.
