Preceded by the discovery of topological insulators, Dirac and Weyl semimetals have become a pivotal direction of research in contemporary condensed matter physics. While easily accessible from a theoretical viewpoint, these topological semimetals pose a serious challenge in terms of experimental synthesis and analysis to allow for their unambiguous identification. In this work, we report on detailed transport experiments on compressively strained HgTe. Due to the superior sample quality in comparison to other topological semimetallic materials, this enables us to resolve the interplay of topological surface states and semimetallic bulk states to an unprecedented degree of precision and complexity. As our gate design allows us to precisely tune the Fermi level at the Weyl and Dirac points, we identify a magnetotransport regime dominated by Weyl/Dirac bulk state conduction for small carrier densities and by topological surface state conduction for larger carrier densities. As such, similar to topological insulators, HgTe provides the archetypical reference for the experimental investigation of topological semimetals.