Ultrafast optical spectroscopy of quantum materials uncovers their intrinsic physical properties such as light-matter interactions and dynamics of particles and quasi-particles. In this presentation, I will present our ultrafast optical studies of two different electronic systems: topological insulators (TIs) and two-dimensional transition metal Dichalcogenides (2D TMDs). In TIs, where Dirac-like topological surface states (TSSs) coexist with an underlying bulk insulator, we have investigated ultrafast dynamics of TSS Dirac fermions and plasmons, and their interactions with phonon by using optical-pump terahertz-probe spectroscopy. We have found that, unlike Dirac electrons in graphene, TSS Dirac electrons exhibit unique dynamic features originating from interactions with coexisting bulk insulator. In the studies of 2D TMDs, we have performed ultrafast optical pump-probe spectroscopy of anisotropic excitons in group-VII TMDs. We discuss coherent light-matter interactions such as excitonic optical Stark effect and quantum beats, both of which exhibit significant laser-polarization dependence, resulting from anisotropic nature of the excitons.