Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive

Question

1. Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive.
2. Prove also that even though the Galois group of $f$ is transitive not every permutation of the roots occur.

It is not difficult to give an example for the second statement e.g. Galois group of $x^4-2$ over $ \mathbb{Q}$, but I don't know how to prove the first statement, any help would be great.