Existing image captioning models are usually trained by cross-entropy (XE) loss and reinforcement learning (RL), which set ground-truth words as hard targets and force the captioning model to learn from them. However, the widely adopted training strategies suffer from misalignment in XE training and inappropriate reward assignment in RL training. To tackle these problems, we introduce a teacher model that serves as a bridge between the ground-truth caption and the caption model by generating some easier-to-learn word proposals as soft targets. The teacher model is constructed by incorporating the ground-truth image attributes into the baseline caption model. To effectively learn from the teacher model, we propose Teacher-Critical Training Strategies (TCTS) for both XE and RL training to facilitate better learning processes for the caption model. Experimental evaluations of several widely adopted caption models on the benchmark MSCOCO dataset show the proposed TCTS comprehensively enhances most evaluation metrics, especially the Bleu and Rouge-L scores, in both training stages. TCTS is able to achieve to-date the best published single model Bleu-4 and Rouge-L performances of 40.2% and 59.4% on the MSCOCO Karpathy test split. Our codes and pre-trained models will be open-sourced.