In recent years, cavity opto-magnonmechanical systems have received much attention, and the coupled-cavity model serves as a classic theoretical framework in this field. This work aims to construct a coupled-cavity opto-magnonmechanical system to study induced transparency, Fano resonance, and fast-slow light effects in such a system. By transferring phenomena typically studied in a single optical cavity to a coupled-cavity system, we analyze specific phenomena detected in optical and microwave cavities, such as transmission and absorption spectra, to investigate induced transparency. We also examine Fano resonance in the model by varying detuning and study fast-slow light effects through group velocity. In this work, first, the corresponding physical model is constructed. Based on the theoretical model, a reasonable Hamiltonian is proposed. By introducing appropriate dissipation and fluctuation noise terms, the Langevin equations of motion are derived. Next, the Langevin equations are linearized, and the resonant terms are retained to obtain the upper band O_+ , which is usually associated with the anti-Stokes process. The amplitude of the field mode is then derived using the input-output relation. Following the experimental data from referenced literature, a numerical simulation program is implemented in Mathematica. By substituting the relevant parameters and performing calculations, the results are obtained through simulation. For the first time, the interactions between photons, magnons, microwaves, and phonons, as well as the interplay between photons in the two cavitie, are investigated in a coupled-cavity opto-magnomechanical system. Electromagnetically induced transparency (EIT), Fano resonance, and fast-slow light effects are studied in this coupled-cavity opto-magnomechanical framework. Phenomena typically examined in a single optical cavity are extended to the coupled-cavity system, with specific observations analyzed separately in the optical and microwave cavities. When the detection field frequency is equal to phonon mode frequency, namely \delta =\omega _b , the absorption spectrum splits, and the absorption peak decreases from its maximum to its minimum. This phenomenon arises from the disruption of quantum interference effects. The resonance condition suppresses the generation of Fano resonance. At the resonant frequency ω₀, the group delay is greater than zero, indicating slow-light propagation, and this effect is enhanced with the increase of coupling strength. Additionally, a group delay τ is achieved. Meanwhile, on either side of the resonant frequency, the group delay peaks exhibit a reducing positive value and an increasing negative value, respectively, signifying the gradual weakening of the slow-light effect and a corresponding enhancement of the fast-light effect. Investigated in this work are the magnetically induced transparency (MIT), magnomechanically induced transparency (MMIT) and opto-mechanically induced transparency (OMIT) windows in a coupled-cavity opto-magnomechanical (OMM) system under a strong control field and weak probe field. The MMIT phenomenon is observed through nonlinear phonon-magnon interactions. Additionally, the photon-magnon interaction in the microwave cavity results in MIT, whereas OMIT is achieved through the radiation pressure interaction between photons and nonlinear phonons in the optical cavity. The frequency of the probe field is tuned to interact with both the microwave and optical cavities. When the probe field is coupled with the microwave cavity, its absorption at the resonant frequency is significantly suppressed under optomechanical coupling, resulting in a significant optical switching effect on transmission. The asymmetric Fano resonance phenomenon, which reflects the existence of quantum interference mechanisms within the system and influences the fast- and slow-light conversion processes is analyzed. Furthermore, appropriate coupling parameters can not only enhance the fast- and slow-light effects but also enable dynamic switching between them.