Where can I find the common names for the zoology taxonomy?
The Tree of Life project has a browse-able tree of all major taxa, but not necessarily all species. You can start at the root of the tree here. Tree of life uses modern phylogenetic nomenclature rather than more traditional Kingdom-Phylum-Class-etc.

Part of the problem that you will run into is that common names differ across regions/countries/languages. This is the point of species binomials. They are language-invariant.