my class notes on the topic are on my web page. look at algebra class notes 843-2.
there are some computations on pages 43-50.
basically, the galois group is based on how different the roots of a polynomial are from each other.
e.g. if whenever you adjoin one root of a polynomial to the previous base field, you never get any other roots for free, then the group is large as possible.
but if when you adjoin just one root, then the other roots are automatically contained in the first extension field, then the group is as small as possible.
e.g. if the polynomial is X^5 - 1, then just adjoining one root, a primitive root of 1, gives all the other roots as powers of that one. hence the field extension of degree 4 obtained by adjoining that one root already has all the roots in it, so the group has order 4.
then you have to figure out which group of order 4 it is.
more precisely, the galois group depends not just on how many other roots you get each time you adjoin another one, but on how the polynomial factors over the various successive adjunction fields.
the group is big as possible if each time you adjoin another root, it only splits off that one linear factor from the polynomial, and the rest remains irreducible.