By now there are more tractable proofs of resolution of singularities than Hironaka's, so it no longer have to be a black box. A relatively elementary approach is described in Kollár's Lectures on Resolution of Singularities. There are many references there to algorithmic resolutions. In particular, in the case of hypersurfaces there is even a Maple program to do it. Check out Bodnár-Schicho's paper Automated Resolution of Singularities for Hypersurfaces and the references in there. (I'm not trying to be comprehensive in this answer to provide all the references, because there are lots of them. But probably most references that are at least 6-8 years old are referenced in the above two sources.)

By now there are more tractable proofs of resolution of singularities than Hironaka's, so it no longer has to be a black box. A relatively elementary approach is described in Kollár's Lectures on Resolution of Singularities. There are many references there to algorithmic resolutions. In particular, in the case of hypersurfaces there is even a Maple program to do it. Check out Bodnár-Schicho's paper Automated Resolution of Singularities for Hypersurfaces and the references in there. (I'm not trying to be comprehensive in this answer to provide all the references, because there are lots of them. But probably most references that are at least 6-8 years old are referenced in the above two sources.)