I am trying to understand the structure of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})\simeq Z_n^*$ for $n \in \mathbb{Z}$ - it would be great if someone could me understand the generators in this group so I can understand the structure
Is it true $\tau (\zeta_n)=\zeta^n$? How can we show the existence of such generators?
Here is a hint, maybe it will be helpful. Ignore the vector space structure of $\Bbb Q(\zeta_n)$ for the moment, and just think of the cyclic group (under complex multiplication) $\{1,\zeta_n,\zeta_n^2,\dots,\zeta_n^{n-1}\}$.
Each of these is a root (in fact they are all of them) of the rational polynomial $x^n - 1$.
Thus an element of $\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q)$ must send such a root to another root, and since they are automorphisms of $\Bbb Q(\zeta_n)$, they induce automorphisms of $\langle \zeta_n\rangle \cong \Bbb Z_n$.
Well, since $\Bbb Z_n$ is cyclic (generated by $1$), to determine an automorphism $\phi$ of $\Bbb Z_n$, it suffices to know where $1$ is mapped to. The corresponding automorphism is then given by:
$k \mapsto ak$, where $a = \phi(1)$.
So ask yourself, which choices of $a \in \Bbb Z_n$ lead to an automorphism $\phi$? Convince yourself $a$ has to be invertible under multiplication modulo $n$, and that this is also sufficient.
Then take the additive structure of $(\Bbb Z_n,+)$ and transfer (via the imaginary exponential map) it to the multiplicative structure of $\langle (\zeta_n\rangle, \cdot)$.
That is, the elements of $\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q)$ you want are of the form $\zeta_n \mapsto (\zeta_n)^a$, with $a$ as in $\phi$ above.