The quick and precise identification of COVID-19 pneumonia, non-COVID-19 viral pneumonia, bacterial pneumonia, mycoplasma pneumonia, and normal lung on chest CT images play a crucial role in timely quarantine and medical treatment. However, manual identification is subject to potential misinterpretations and time-consumption issues owing the visual similarities of pneumonia lesions. In this study, we propose a novel multi-scale attention network (MSANet) based on a bag of advanced deep learning techniques for the automatic classification of COVID-19 and multiple types of pneumonia. The proposed method can automatically pay attention to discriminative information and multi-scale features of pneumonia lesions for better classification. The experimental results show that the proposed MSANet can achieve an overall precision of 97.31%, recall of 96.18%, F1-score of 96.71%, accuracy of 97.46%, and macro-average area under the receiver operating characteristic curve (AUC) of 0.9981 to distinguish between multiple classes of pneumonia. These promising results indicate that the proposed method can significantly assist physicians and radiologists in medical diagnosis.