Glaucoma, a leading cause of permanent blindness worldwide, necessitates early detection to prevent vision loss, a task that is challenging and time-consuming when performed manually. This study proposes an automatic glaucoma detection method on enhanced retinal images using deep learning. The system analyzes retinal images, generating masks for the optic disc and optic cup, and providing a classification for glaucoma diagnosis. We employ a U-Net architecture with a pretrained residual neural network (ResNet34) for segmentation and an EfficientNetB0 for classification. The proposed framework is tested on publicly available datasets, including ORIGA, REFUGE, RIM-ONE DL, and HRF. Our work evaluated the U-Net model with five pretrained backbones (ResNet34, ResNet50, VGG19, DenseNet121, and EfficientNetB0) and examined preprocessing effects. We optimized model training with limited data using transfer learning and data augmentation techniques. The segmentation model achieves a mean intersection over union (mIoU) value of 0.98. The classification model shows remarkable performance with 99.9% training and 100% testing accuracy on ORIGA, 99.9% training and 99% testing accuracy on RIM-ONE DL, and 98% training and 100% testing accuracy on HRF. The proposed model outperforms related works and demonstrates potential for accurate glaucoma classification and detection tasks.