Amount of generators for cyclic group and Euler's function

A cyclic group with $n$ elements has $\varphi(n)$ generators.

It seems like what you're getting confused by is that:

• The group $\mathbb{Z}_7^\times=\{1,2,3,4,5,6\}$ is a cyclic group with 6 elements, and therefore it should considered equivalent to (more precisely, it is isomorphic to) the cyclic group $\mathbb{Z}_6$, and you should expect $\varphi(6)=2$ generators, not $\varphi(7)=6$ generators

• The elements $4$ and $6$ in $\mathbb{Z}_7^\times$ are not generators: the powers of $4$ in $\mathbb{Z}_7^\times$ are $\{1,4,2\}$, and the powers of $6$ in $\mathbb{Z}_7^\times$ are $\{1,6\}$. (The only generators of $\mathbb{Z}_7^\times$ are $3$ and $5$, and there are exactly $2=\varphi(6)$ of them.)

More generally, if $p$ is a prime number, then $\mathbb{Z}_p$ is a cyclic group with $p$ elements, while $\mathbb{Z}_p^\times$ is a cyclic group with $p-1$ elements, so $\mathbb{Z}_p^\times$ has $\varphi(p-1)$ generators.

As for the best method to find a generator, that is a far more advanced problem.