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 propose the use of a modified minimum mean squared error (MMSE) shrinkage function in Trainable Iterative Soft Thresholding Algorithm (TISTA). Then, we propose to learn this MMSE shrinkage function using a model-based neural network. In the simulation results, the proposed learning-based method L-MMSE-MMV-TISTA gives a 30-40% reduction in preamble requirement is seen compared to TISTA. The proposed method is also compared with a MMV sparse Bayesian learning (M-SBL) method. While M-SBL can provide better performance at the cost of higher complexity in some measurement-constrained settings, LMMSE-MMV-TISTA provides a significant complexity advantage.