I'm wondering how to go about proving that certain ideals are principal or prime or maximal. I don't have a grasp of the general method if I'm given an ideal in a ring and told to prove it is maximal or prime, or if I'm given a ring and told to prove that it is (or isn't) a PID.

For example:

Prove (3,x) is maximal in Z[x]. Is (2,x) maximal? (5,x)?

Prove that (3) and (x) are prime ideals in Z[x].

Is Z[x] a PID? I understand why Z is a PID but can't generalize past it.

If anyone can also show me some more esoteric examples (since these are pretty standard ones), that'd be great. Thanks!