Objective. Early detection and diagnosis allow for intervention and treatment at an early stage of breast cancer. Despite recent advances in computer aided diagnosis systems based on convolutional neural networks for breast cancer diagnosis, improving the classification performance of mammograms remains a challenge due to the various sizes of breast lesions and difficult extraction of small lesion features. To obtain more accurate classification results, many studies choose to directly classify region of interest (ROI) annotations, but labeling ROIs is labor intensive. The purpose of this research is to design a novel network to automatically classify mammogram image as cancer and no cancer, aiming to mitigate or address the above challenges and help radiologists perform mammogram diagnosis more accurately. Approach. We propose a novel feature selection and enhancement network (FSE-Net) to fully exploit the features of mammogram images, which requires only mammogram images and image-level labels without any bounding boxes or masks. Specifically, to obtain more contextual information, an effective feature selection module is proposed to adaptively select the receptive fields and fuse features from receptive fields of different scales. Moreover, a feature enhancement module is designed to explore the correlation between feature maps of different resolutions and to enhance the representation capacity of low-resolution feature maps with high-resolution feature maps. Main results. The performance of the proposed network has been evaluated on the CBIS-DDSM dataset and INbreast dataset. It achieves an accuracy of 0.806 with an AUC of 0.866 on the CBIS-DDSM dataset and an accuracy of 0.956 with an AUC of 0.974 on the INbreast dataset. Significance. Through extensive experiments and saliency map visualization analysis, the proposed network achieves the satisfactory performance in the mammogram classification task, and can roughly locate suspicious regions to assist in the final prediction of the entire images.