The combination of non-orthogonal multiple access (NOMA) and reconfigurable intelligent surface (RIS) is capable of boosting spectrum and energy efficiency for future communication networks. In this letter, we aim to minimize the total transmit power for uplink multi-group RIS-aided NOMA system via jointly designing the transceivers and phase shifts. Particularly, a novel framework with group-based successive interference cancellation (GSIC) is conceived, wherein the user groups associated with different RISs are served by NOMA and the GSIC is invoked to separate user signals in different groups. Furthermore, to facilitate the process, the original optimization problem is decomposed into two subproblems, i.e., transceiver design and phase shift optimization. In the transceiver design, a parallel iteration algorithm is developed to obtain the optimal transmit power when the minimum mean square error receiver is employed. For the optimization of phase shifts, a sequential rotation scheme is proposed to properly devise each phase shifter. Simulation results show that the proposed method outperforms various benchmarks in terms of total transmit power.