Number of Elements of order $p$ in $S_{p}$

Maybe you mean the number of elements of order dividing $p$ (so that you are including the identity)? (Think about the case $p = 3$ --- there are two three cycles, not three of them.) For the general question, think about the possible cycle structure of an element of order $p$ in $S_p$.

You can go from the formula in your question to Wilson's theorem by counting the number of $p$-Sylow subgroups (each contains $p-1$ elements of order $p$), and then appealing to Sylow's theorem. (You will find that there are $(p-2)!$ $p$-Sylow subgroups, and by Sylow's theorem this number is congruent to $1$ mod $p$. Multiplying by $p-1$, we find that $(p-1)!$ is congruent to $-1$ mod $p$.)

Every element of order $p$ in $S_p$ is a $p$-cycle. The symmetric group $S_{p-1}$ acts transitively on these $p$ cycles.