Recent interest in developing fast spintronic devices and laser-controllable magnetic solids has sparked tremendous experimental and theoretical efforts to understand and manipulate ultrafast dynamics in materials. Studies of spin dynamics in the terahertz (THz) frequency range are particularly important for elucidating microscopic pathways toward novel device functionalities. Here, we review THz phenomena related to spin dynamics in rare-earth orthoferrites, a class of materials promising for antiferromagnetic spintronics. We expand this topic into a description of four key elements. (1) We start by describing THz spectroscopy of spin excitations for probing magnetic phase transitions in thermal equilibrium. While acoustic magnons are useful indicators of spin reorientation transitions, electromagnons that arise from dynamic magnetoelectric couplings serve as a signature of inversion-symmetry-breaking phases at low temperatures. (2) We then review the strong laser driving scenario, where the system is excited far from equilibrium and thereby subject to modifications to the free-energy landscape. Microscopic pathways for ultrafast laser manipulation of magnetic order are discussed. (3) Furthermore, we review a variety of protocols to manipulate coherent THz magnons in time and space, which are useful capabilities for antiferromagnetic spintronic applications. (4) Finally, new insights into the connection between dynamic magnetic coupling in condensed matter and the Dicke superradiant phase transition in quantum optics are provided. By presenting a review on an array of THz spin phenomena occurring in a single class of materials, we hope to trigger interdisciplinary efforts that actively seek connections between subfields of spintronics, which will facilitate the invention of new protocols of active spin control and quantum phase engineering.