The symmetric group on $n$ symbols is not cyclic when $n>2$ (it's not even abelian). Hence it cannot be generated by a single permutation. However, it can always be generated by two permutations, a transposition $(12)$ and a cycle $(123\ldots n)$.
I will assume permutations act on the right. Throughout $i,j$ will refer to any integers with $1\le i,j\le n$.
Let $\sigma,\omega\in S_n$. Denote $\sigma^\omega=\omega^{-1}\sigma\omega$ and notice $(i^\omega)^{(\sigma^\omega)}=(i^\sigma)^\omega$ so if, for example, $\sigma=(x_1,\ldots,x_r)$ is a cycle then $\sigma^\omega=(x_1^\omega,\ldots,x_r^\omega)$.
Write $\sigma_1=(1,2,\ldots,n)$ and $\sigma_2=(1,2,\ldots, n-1)$. I will answer both interpretations of the question, so I will calculate $C_{S_n}(\sigma_1)$, $C_{S_n}(\sigma_2)$ and $C_{S_n}(\{\sigma_1,\sigma_2\})=C_{S_n}(\sigma_1)\cap C_{S_n}(\sigma_2)$.
Suppose $\omega\in C_{S_n}(\sigma_1)$, so $\sigma_1^\omega=\sigma_1$. From the above, $\sigma_1=\sigma_1^\omega=(1^\omega,\ldots,n^\omega)$. Say $1\le i\le n-1$, then $i^{\sigma_1}=i+1$ and $n^{\sigma_1}=1$. In particular if $i^\omega=j$ then (modulo $n$) $(i+1)^\omega=j+1=i^\omega+1$. Hence $\omega\in\langle \sigma_1\rangle$, but $C_{S_n}(\sigma_1)\supseteq\langle \sigma_1\rangle$ so $C_{S_n}(\sigma_1)=\langle \sigma_1\rangle$.
Suppose $\omega\in C_{S_n}(\sigma_2)$, then by the above $(n^\omega)^{\sigma_2}=(n^\omega)^{(\sigma_2^\omega)}=(n^{\sigma_2})^\omega=n^\omega$. The only fixed point of $\sigma_2$ is $n$ so $n^\omega=n$. Therefore $\omega\in S_{n-1}$ so we are reduced to the $\sigma_1$ case (but for $n-1$) so $C_{S_n}(\sigma_2)=\langle \sigma_2\rangle$.
Finally we have $C_{S_n}(\{\sigma_1,\sigma_2\})=C_{S_n}(\sigma_1)\cap C_{S_n}(\sigma_2)=\{1_{S_n}\}$
Best Answer
Two permutations are disjoint if any point moved by one permutation is fixed by the other permutation. In other words, in the disjoint cycle decomposition of the two permutations, there is no overlap in the points that are written out. Equivalently, two permutations are disjoint if and only if they have disjoint support (the support of a permutation is the set of points moved by the permutation).
For example, in $S_9$, the permutations $(12)(34)$ and $(569)$ are disjoint permutations because the sets $\{1,2,3,4\}$ and $\{5,6,9\}$ are disjoint. The permutations $(12)(34)$ and $(561)$ are not disjoint.