Diabetic retinopathy (DR) is the leading cause of vision loss for adult individuals, and early fundus screening can significantly reduce this visual loss. Color fundus image is often used in large-scale fundus screening due to the acquisition convenience and its human-harmless. As a kind of red lesions in fundus images, the appearance of microaneurysms is the main marker of mild non-proliferative DR, and hemorrhage, as another kind of red lesions, is related to moderate and severe non-proliferative DR. So that red lesions in fundus images are important indicators for the screening of DR. This study proposes a multi-task network, named Red-Seg, for red lesion segmentation. The network contains two individual branches, each is used for one kind of lesion segmentation. Meantime, a two-stage training algorithm is presented where different loss functions are used in different stages. In the first stage, modified Top-k balanced cross-entropy loss is used to push the network focuses on hard-to-classify samples. And, in the second stage, false positive and false negative are integrated as loss function into training to reduce misclassification further. At last, extensive experiments are employed on the IDRiD dataset, and the lesion segmentation results are compared with other methods. Experimental results show that proposed two-stage training algorithm can lead to much higher precision and recall, which means this method can reduce misclassification in some certain. Specifically for hemorrhage segmentation, both recall and precision increased by at least 2.8%. Meanwhile, compared with other image-level lesion segmentation models, such as HED, FCRN, DeepLabv3+, and L-Seg, Red-Seg achieves much higher AUC_PR on microaneurysm segmentation.