ag.algebraic geometry - Reference for explicit calculation of blowups (of ideals) and strict transforms

I think, you'd like to see explicit equations and if possible, a singularity completely solved by blowups. One can take a look at such thinks in Lectures on resolution of singularities by Janos Kollar (http://press.princeton.edu/titles/8449.html). The book starts with more than ten different methods of how to solve singularities on curves and then continues on resolution of singularities on surfaces. There you will see specific examples and explicit equations.