Galois groups of polynomials and explicit equations for the roots

One of the fundamental techniques for doing this is to compute the (Lagrange) resolvants associated to the equation. Assuming the group is solvable, the resolvants will be factorizable and amenable to lower degree auxiliary equations. To have a first idea of all this, I would advice you to look at Harold Edwards splendid book : "Galois Theory" (ISBN 038790980X) where it does it for the elementary cases and an example of the cyclotomic equation.

With modern Computer Algebra System, one can more easily explore the idea further, but considerable ingenuity is needed because the degree of the auxiliary equations rise quickly. Some CAS (such as GAP which is a free and open-source academic research tool) are able to give you the Galois groups of low degree polynomials, as well as properties of splitting fields and you can look at their tutorial to have examples of their use.

Effective and Inverse Galois Theory is still an active research subject. Some of the works by Annick Valibouze, N. Durov, Klueners, etc. might help you to go further.

The book Classical Galois theory: with examples by Gaal contains explicit computations, if not a complete algorithm.

I feel the book "Algebraic theories" by Dickson(now its coming in Dover phoenix edition) have some "Real" stuff about Galois theory in the sense that answers your question of explicit calculations.