In this paper, we propose a novel progressive learning (PL) framework for deep neural network (DNN) based speech enhancement. It aims at decomposing the complicated regression problem of mapping noisy to clean speech into a series of subproblems for enhancing system performances and reducing model complexities. As an illustration, we design a signal-to-noise ratio (SNR) based PL architecture by guiding each hidden layer of the DNN to learn an intermediate target with gradual SNR gains explicitly. Furthermore, post-processing, with the rich set of information from the multiple learning targets, can further be conducted. Experimental results demonstrate that SNR-based progressive learning can effectively improve perceptual evaluation of speech quality and short-time objective intelligibility in low SNR environments, and reduce the model parameters by 50% when compared with the DNN baseline system. Moreover, when combined with post-processing, the proposed approach can be further improved.