We study the dynamics of a three-level system (ThLS) sinusoidally driven in both longitudinal and transverse directions and in the presence of a uniaxial anisotropy $D$ entering the generic Hamiltonian through the zero-energy splitting term $D(S^{z})^{2}$ where $S^{z}$ is the projection of the spin vector along the quantization direction. As a consequence of the addition of this term, the order of the symmetry group of the Hamiltonian is increased by a unit and we observe a sequence of cascaded $SU(3)$ Landau-Zener-St��ckelberg-Majorana (LZSM) interferometers. The study is carried out by analytically and numerically calculating the probabilities of non-adiabatic and adiabatic evolutions. For non-adiabatic evolutions, two main approximations based on the weak and strong driving limits are discussed by comparing the characteristic frequency of the longitudinal drive with the amplitudes of driven fields. For each of the cases discussed, our analytical results quite well reproduce the gross temporal profile of the exact numerical probabilities. This allows us to check the range of validity of analytical results and confirm our assumptions. For adiabatic evolutions, a general theory is constructed allowing for the description of adiabatic passages in arbitrary ThLSs in which direct transitions between states with extremal spin projections are forbidden. A compact formula for adiabatic evolutions is derived and numerically tested for some illustrative cases. Interference patterns demonstrating multiple LZSM transitions are reported. Applications of our results to the Nitrogen Vacancy Center (NVC) in diamond are discussed.