Movable antenna (MA) is an emerging technology that utilizes localized antenna movement to achieve better channel conditions for enhancing communication performance. In this paper, we study the MA-enhanced multicast transmission from a base station equipped with multiple MAs to multiple groups of single-MA users. Our goal is to maximize the minimum weighted signal-to-interference-plus-noise ratio (SINR) among all the users by jointly optimizing the position of each transmit/receive MA and the transmit beamforming. To tackle this challenging problem, we first consider the single-group scenario and propose an efficient algorithm based on the techniques of alternating optimization and successive convex approximation. Particularly, when optimizing transmit or receive MA positions, we construct a concave lower bound for the signal-to-noise ratio (SNR) of each user using only the second-order Taylor expansion, which simplifies the problem-solving process compared to the existing two-step approximation method. The proposed design is then extended to the general multi-group scenario. Simulation results show that the proposed algorithm converges faster than the existing two-step approximation method, achieving a 3.4% enhancement in max-min SNR. Moreover, it can improve the max-min SNR/SINR by up to 22.5%, 181.7%, and 343.9% compared to benchmarks employing only receive MAs, only transmit MAs, and both transmit and receive FPAs, respectively.