We consider the problem of joint activity detection and channel estimation in massive random access. When the receiver has multiple antennas, this is a joint sparse recovery problem with multiple measurement vectors (MMV). For the general setting where the channels could be correlated across antennas, we first develop a modified minimum mean squared error (MMSE) shrinkage function to be used in the Trainable Iterative Soft Thresholding Algorithm (TISTA). Then, we learn this MMSE shrinkage function using a model-based neural network. In the simulation results, the proposed learning-based method, L-MMSE-MMV-TISTA, offers a 30-40% reduction in preamble length requirement compared to TISTA. We also compare L-MMSE-MMV-TISTA with the state-of-the-art MMV sparse Bayesian learning (M-SBL) method. While M-SBL can provide better performance at the cost of higher complexity in highly measurement-constrained settings, LMMSE-MMV-TISTA provides a significant complexity advantage when only a slightly larger number of measurements are available.