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We study how the information flows in many-body dynamics governed by random quantum circuits and discover a rich set of dynamical phase transitions in this information flow. The phase-transition points and their critical exponents are established across Clifford and Haar random circuits through finite-size scaling. The flow of both classical and quantum information, measured respectively by Holevo and coherent information, shows similar dynamical phase transition behaviors. We investigate how the phase transitions depend on the initial location of the information and the final probe region, and find ubiquitous behaviors in these transitions, revealing interesting properties about the information propagation and scrambling in this quantum many-body model. Our paper underscores rich behaviors of the information flow in large systems with numerous phase transitions, thereby sheds light on the understanding of quantum many-body dynamics.