After photoexcitation, molecules can follow many different paths for electronic relaxation. Of these paths, electronic transitions between states of different spin-multiplicities, intersystem crossings, have largely been ignored in many systems because of their spin-forbidden nature, but recent work has shown that intersystem crossings can occur at short times and compete with other relaxation pathways. Understanding these complicated excited state processes requires both advanced excited state dynamics algorithms and excited state electronic structure. In the first part of the work presented here, we investigate a variety of inexpensive, mixed quantum-classical approaches for nonadiabatic dynamics. We find that modern approaches have made great advances but may lead to instabilities when unphysical trajectories are included. In the second part of this work, methods are developed for the requisite electronic structure for nonadiabatic dynamics with spin-orbit coupling. We present a computationally efficient algorithm for computing the energies, analytic gradients and nonadiabatic derivative couplings of spin-adiabatic states corresponding to configuration interaction singles or time-dependent density functional theory incorporating spin-orbit couplings. We expect this algorithm will be useful for future exploration of intersystem crossings in real molecular systems.