Using first-principles density functional theory calculations, combined with a topological analysis, we have investigated the electronic properties of ${\text{Cd}}_{3}{\text{As}}_{2}$ and ${\text{Na}}_{3}\text{Bi}$ Dirac topological semimetals doped with nonmagnetic and magnetic impurities. Our systematic analysis shows that the selective breaking of the inversion, rotational, and time-reversal symmetry, controlled by specific choices of the impurity doping, induces phase transitions from the original Dirac semimetal to a variety of topological phases such as topological insulator, trivial semimetal, nonmagnetic and magnetic Weyl semimetal, and Chern insulator. The Dirac semimetal phase can exist only if the rotational symmetry ${C}_{n}$ with $n>2$ is maintained. One particularly interesting phase emerging in doped ${\text{Cd}}_{3}{\text{As}}_{2}$ is a coexisting Dirac-Weyl phase, which occurs when only inversion symmetry is broken while time-reversal symmetry and rotational symmetry are both preserved. To further characterize the low-energy excitations of this phase, we have complemented our density functional results with a continuum four-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model, which indeed displays nodal points of both Dirac and Weyl types. The coexisting phase appears as a transition point between two topologically distinct Dirac phases but may also survive in a small region of parameter space controlled by external strain.