The combination of magnetic symmetries and electronic band topology provides a promising route for realizing topologically nontrivial quasiparticles, and the manipulation of magnetic structures may enable the switching between topological phases, with the potential for achieving functional physical properties. Here, we report measurements of the electrical resistivity of EuCd$_2$As$_2$ under pressure, which show an intriguing insulating dome at pressures between $p_{\rm c1}\sim1.0$~GPa and $p_{\rm c2}\sim2.0$~GPa, situated between two regimes with metallic transport. The insulating state can be fully suppressed by a small magnetic field, leading to a colossal negative magnetoresistance on the order of $10^5$\%, accessible via a modest field of $\sim0.2$~T. First-principles calculations reveal that the dramatic evolution of the resistivity under pressure is due to consecutive transitions of EuCd$_2$As$_2$ from a magnetic topological insulator to a trivial insulator, and then to a Weyl semimetal, with the latter resulting from a pressure-induced change in the magnetic ground state. Similarly, the colossal magnetoresistance results from a field-induced polarization of the magnetic moments, transforming EuCd$_2$As$_2$ from a trivial insulator to a Weyl semimetal. These findings underscore weak magnetic exchange couplings and spin anisotropy as ingredients for discovering tunable magnetic topological materials with desirable functionalities.