With the recent outbreak of COVID-19, fast diagnostic testing has become one of the major challenges due to the critical shortage of test kit. Pneumonia, a major effect of COVID-19, needs to be urgently diagnosed along with its underlying reasons. In this paper, deep learning aided automated COVID-19 and other pneumonia detection schemes are proposed utilizing a small amount of COVID-19 chest X-rays. A deep convolutional neural network (CNN) based architecture, named as CovXNet, is proposed that utilizes depthwise convolution with varying dilation rates for efficiently extracting diversified features from chest X-rays. Since the chest X-ray images corresponding to COVID-19 caused pneumonia and other traditional pneumonias have significant similarities, at first, a large number of chest X-rays corresponding to normal and (viral/bacterial) pneumonia patients are used to train the proposed CovXNet. Learning of this initial training phase is transferred with some additional fine-tuning layers that are further trained with a smaller number of chest X-rays corresponding to COVID-19 and other pneumonia patients. In the proposed method, different forms of CovXNets are designed and trained with X-ray images of various resolutions and for further optimization of their predictions, a stacking algorithm is employed. Finally, a gradient-based discriminative localization is integrated to distinguish the abnormal regions of X-ray images referring to different types of pneumonia. Extensive experimentations using two different datasets provide very satisfactory detection performance with accuracy of 97.4% for COVID/Normal, 96.9% for COVID/Viral pneumonia, 94.7% for COVID/Bacterial pneumonia, and 90.2% for multiclass COVID/normal/Viral/Bacterial pneumonias. Hence, the proposed schemes can serve as an efficient tool in the current state of COVID-19 pandemic. All the architectures are made publicly available at: https://github.com/Perceptron21/CovXNet.