A flexible-grid $1\times (2\times 3)$ mode- and wavelength-selective switch which comprises counter-tapered couplers and silicon microring resonators has been proposed, optimized, and demonstrated experimentally in this work. By carefully thermally tuning phase shifters and silicon microring resonators, mode and wavelength signals can be independently and flexibly conveyed to any one of the output ports, and different bandwidths can be generated as desired. The particle swarm optimization algorithm and finite difference time-domain method are employed to optimize structural parameters of the two-mode (de)multiplexer and crossing waveguide. The bandwidth-tunable wavelength-selective optical router composed of 12 microring resonators is studied by taking advantage of the transfer matrix method. Measurement results show that, for the fabricated module, cross talk less than $-10.18\mathrm{dB}$, an extinction ratio larger than 17.41 dB, an in-band ripple lower than 0.79 dB, and a 3-dB bandwidth changing from 0.38 to 1.05 nm are obtained, as the wavelength-channel spacing is 0.40 nm. The corresponding response time is measured to be 13.64 µs.