Training beam design for channel estimation with infinite-resolution and low-resolution phase shifters (PSs) in hybrid analog-digital milimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems is considered in this paper. By exploiting the sparsity of mmWave channels, the optimization of the sensing matrices (corresponding to training beams) is formulated according to the compressive sensing (CS) theory. Under the condition of infinite-resolution PSs, we propose relevant algorithms to construct the sensing matrix, where the theory of convex optimization and the gradient descent in Riemannian manifold is used to design the digital and analog part, respectively. Furthermore, a block-wise alternating hybrid analog-digital algorithm is proposed to tackle the design of training beams with low-resolution PSs, where the performance degeneration caused by non-convex constant modulus and discrete phase constraints is effectively compensated to some extent thanks to the iterations among blocks. Finally, the orthogonal matching pursuit (OMP) based estimator is adopted for achieving an effective recovery of the sparse mmWave channel. Simulation results demonstrate the performance advantages of proposed algorithms compared with some existing schemes.