In the present paper, we propose and experimentally verify a concept of a magnonic reservoir computer. The system utilizes the nonlinear behavior of propagating magnetostatic surface spin waves in a yttrium-iron garnet thin film and the time delay inherent in the active ring configuration to process time-dependent data streams. Higher reservoir dimensionality is obtained through the time-multiplexing method, whereby inputs to the system are multiplied by a mask to drive complex dynamics in the ring and the output is sampled in time to create a series of "virtual"neurons for each sample. The input mask is implemented as a train of microwave pulses injected to the system. To demonstrate the efficacy of the concept, the reservoir computer is evaluated on the short-term memory and parity-check benchmark tasks, and the physical system parameters are tuned to optimize performance. By incorporating a reference line to mix the input signal directly onto the ring-resonator output, both the amplitude and phase nonlinearity of the spin waves can be exploited. The addition of a second spin-wave delay line increases the delay time of the ring and enhances the fading memory capacity of the system. Configuring the second delay line to transmit backward volume spin waves also partly compensates the dispersive pulse broadening that is present because of the large delay time.