In this paper, a dual-band terahertz metamaterial sensor based on aluminum and silicon is proposed and simulated. The aluminum surface, which is deposited on a silicon substrate, is made of a C-shaped frame resonator, a rectangular beam, and a cross. The device is insensitive to the change of incident angle in the range of 0°-30°, which shows the great transmission stability of the sensor. By examining the resonance frequency shift, it is shown that 98.3 and 237.5 GHz/RIU refractive index sensitivity can be obtained near 1.76 and 2.404 THz transmission dips of the proposed structure, respectively. The two dips can be used to sense analytes in different refractive index ranges, respectively. For Dip 1 at 1.76 THz, the range is 1.0-1.6. For Dip 2 at 2.404 THz, the range is 1.6-2.0. Different from traditional multi-band metamaterial sensors, two dips generated by the proposed device can measure continuous and non-multiplexed refractive index ranges, respectively. Because the resonance frequencies of matters are different, such a characteristic enables the device to measure different types of analyte using the appropriate resonant peak. A central-relief design is then proposed based on perturbation theory to further improve its sensing performance. The aluminum cross is covered by polyimide, which can interfere with the scattering field on the metal surface and affect the transmission results. For both transmission dips, the optimized structure realizes higher sensitivities of 111.7 GHz/RIU and 262.5 GHz/RIU, respectively. More significantly, the optimized structure also has the characteristic of a wide and non-multiplexed refractive index range. In addition, the effects of analyte thickness and polyimide layer thickness on sensor performance are also discussed. The proposed structure opens up new prospects in the design of multiple-band terahertz metamaterial sensors. It can also meet the sensing needs of biomedical, environmental monitoring, and industrial manufacturing.