So $p$ is prime, and we have $f = x^p -1 \in \mathbb{Q}[x]$ with splitting field $E$. I need to show that ${\rm Gal}(E/\mathbb{Q})$ is abelian and of order $p-1$.
The splitting field $E$ is $\mathbb{Q}(\zeta)$ for some primitive $p$th root of unity. Let $\zeta = e^{\pi i/p}$. From the theory of cyclotomic polynomials, the field $E$ is the splitting field of $f$, and also $[E:\mathbb{Q}] = \phi(p) = p-1 = \left|{\rm Gal}(E/\mathbb{Q})\right|$. Since any $\sigma$ in the Galois group moves $\zeta$ to another $p$th root of unity, every such $\sigma$ is determined by its image $\sigma(\zeta)$, correct? I don't really know how to proceed from here to show that the Galois group is abelian.
Yes, you are correct when saying "every such $\sigma$ is determined by its image $\sigma(\zeta)$".
What do you know about $\sigma(\zeta)$ ? For instance, what is $\sigma(\zeta)^p$ ? Can you have $\sigma(\zeta)^k = 1$ if $1<k<p$ ? You can prove actually that $\sigma(\zeta)$ is a primitive root of the unity, so that $\sigma(\zeta) = \zeta^{k_{\sigma}}$ for some integer $k_{\sigma}$ which is coprime with $p$. Therefore, we can assume $1 \leq k_{\sigma} \leq p-1$.
Therefore you can consider the map $h : \text{Gal}(E/\Bbb Q) \longrightarrow (\Bbb Z/p\Bbb Z)^*$ defined by $\sigma \longmapsto [k_{\sigma}]_p$.
Then you can check that $h$ is a group isomorphism. In particular, the Galois group $\text{Gal}(E/\Bbb Q)$ of the Galois extension $E=\Bbb Q(\zeta_p)/\Bbb Q$ is abelian.
This can be generalized to compute the Galois group of $\Bbb Q(\zeta_n)/\Bbb Q$, where $n$ is composite. See for instance here for more details.
Moreover, if a Galois extension $F/\Bbb Q$ has abelian Galois group, then $F$ can be embedded in a cyclotomic extension $\Bbb Q(\zeta_n)$ ! This is known as the Kronecker-Weber theorem.