In this paper, a metal–insulator–metal (MIM) type surface plasmon waveguide structure is designed, which is composed of a circular split-ring resonance cavity (CSRRC) and a double symmetric rectangular stub waveguide (DSRSW). The finite element method (FEM) is used to investigate the transmission characteristics of the structure. The results show that the interference between the wide-band continuous state excited by the DSRSW and the narrow-band discrete state excited by the CSRRC produces double Fano resonances. The resonance wavelength and line shape of the Fano resonance can be tuned by the geometric parameters of the structure. After optimizing the geometric parameters, the sensitivity of the structure can be up to 1180 nm/RIU, and the figure of merit (FOM) can be up to 5585.3. Meanwhile, the structure can produce a maximum optical delay of about 0.128 ps, and the corresponding group refractive index is about 42.67. This structure has potential applications in refractive index sensors and slow light devices.