Is every symmetric group generated by some cycles?

Question

Actually, I have two questions here:

1. How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details?
2. Is every symmetric group generated by some cycles?

For question (1), I know that, apart from considering the symmetry of triangle, I can also define the isomorphism explicitly, namely, $\phi:D_3 \rightarrow S_3$, $a \mapsto [1,2,3]$, $b \mapsto [1,2]$, where $a$ and $b$ are the generators of $D_3$. It all makes sense to me but I find it hard to prove that $\phi$ is actually a homomorphism.

I come up with question (2) by observing the solution of question (1). In question (1), we make use of the fact that $S_3$ is generated by $[1,2,3]$ and $[1,2]$. But is it true that, for all $n \in \mathbb{N}$, $S_n$ is generated by some cycles?

Thanks in advance.