We present magnetic-field-dependent evolution of magnetization and spin dynamics in the quasi-two-dimensional spin-1/2 magnet Cu₂(OH)₃Br, consisting of alternately coupled ferromagnetic Cu1 and antiferromagnetic Cu2 spin chains. Terahertz spectroscopy reveals a pronounced field-direction dependence of the low-energy magnetic excitation spectrum. For magnetic fields applied perpendicular to the spin chains, B∥ a and B∥ c^*, the spectra undergo abrupt reconstructions at the spin-flop transitions identified independently by high-field magnetization measurements. For B∥ b, by contrast, no spin-flop occurs; instead, the excitation spectrum evolves continuously with field and exhibits a strong terahertz radiation polarization dependence as the ferromagnetic Cu1 subsystem becomes progressively polarized. At higher fields, the complex low-field spectrum is replaced by a reduced set of broad excitations, consistent with a weakening of the coupling between the ferromagnetic and antiferromagnetic chain subsystems. Complementary Raman spectroscopy resolves magnon and spinon excitations alongside several phonon modes and traces the characteristic temperature- and magnetic-field-dependent evolution of the magnetic excitations. The combined spectroscopic and magnetization results map out how a magnetic field reorganizes the coupled ferromagnetic and antiferromagnetic subsystems in Cu₂(OH)₃Br across field-induced phase transitions.