We consider the problem of millimeter-wave (mmWave) channel estimation with a hybrid digital-analog two-stage beamforming structure. A radio frequency (RF) chain excites a dedicated set of antenna subarrays. To compensate for the severe path loss, known training signals are beamformed and swept to scan the angular space. Since the mmWave channels typically exhibit sparsity, the channel response can usually be expressed as a linear combination of a small number of scattering clusters. Thereby the number of angles of arrival (AoAs) and angles of departure (AoDs) with significant signal components is limited, and compressive sensing techniques can be leveraged for estimating the channel. In this paper, we investigate two sparse recovery algorithms: a Bayesian and non-Bayesian one. In the Bayesian approach, we invoke the sparse Bayesian learning (SBL) framework, which relies on a 2-layer hierarchical prior model for channel. A highly efficient and fast iterative Bayesian inference method is then applied to the proposed model. The non-Bayesian approach is a LASSO-based approach, where we devise a low complexity solution by adopting alternating directions method of multipliers (ADMM) technique to solve the problem. The efficacy of the proposed algorithms is demonstrated using numerical examples. The Bayesian approach shows improved estimation performance in relation to the non-Bayesian approach.