This paper considers downlink multi-group multicasting via beamforming facilitated by a reconfigurable intelligent surface (RIS). We develop a fast and scalable algorithm for the joint base station (BS) and RIS beamforming optimization to minimize the transmit power while meeting user quality-of-service (QoS) targets. By analyzing the structure of the QoS constraints, we reformulate the problem and show that the joint beamforming optimization inherently consists of a multicast beamforming QoS problem for the BS and a passive multicast beamforming max-min-fair (MMF) problem for the RIS. We propose a fast alternating multicast beamforming (AMBF) algorithm to effectively solve the two subproblems alternatingly. For the BS multicast subproblem, we utilize the optimal multicast beamforming structure to efficiently determine the BS beamformers. For the RIS multicast subproblem, we reformulate the MMF problem and apply a first-order projected subgradient algorithm (PSA), which yields simple closed-form updates. The computational complexity of the AMBF algorithm grows linearly with the number of RIS elements and BS antennas. We further consider joint BS and RIS beamforming for the weighted MMF design objective, subject to the BS transmit power budget. We propose an alternating PSA (APSA) fast algorithm to compute the beamforming solutions for the BS and RIS. APSA consists of only closed-form updates per iteration, yielding linear computational complexity in the number of RIS elements and BS antennas. Simulation results show the efficacy of our proposed algorithms in terms of performance and computational cost compared to alternative methods.