A metal-insulator-metal waveguide structure composed of a hexagonal resonator cavity and a ring with a slit is proposed. By using the finite difference time domain method, the transmission properties of the structure were studied. It was found that three distinct plasmon-induced transparency peaks appear in the visible and near-infrared bands, and the transmissivity of the three peaks is more than 80%. By tuning the structure size, the positions of the peaks can be adjusted. Then we introduced graphene, covering the surface of the cavity. By adjusting the refraction index of the graphene using light, the position of the three transmission peaks can be changed correspondingly. Based on the effect, we designed an all-optical switcher with ultrafast optical response time (about 2 ps) and low light absorption (about 2.3%). The proposed waveguide structure provides a way for the development of visible and near-infrared filters and all-optical switchers.