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Analyzing the morphological attributes of blood vessels plays a critical role in the computer-aided diagnosis of many cardiovascular and ophthalmologic diseases. Although being extensively studied, segmentation of blood vessels, particularly thin vessels and capillaries, remains challenging mainly due to the lack of an effective interaction between local and global features. In this paper, we propose a novel deep learning model called PC-Net to segment retinal vessels and major arteries in 2D fundus image and 3D computed tomography angiography (CTA) scans, respectively. In PC-Net, the pyramid squeeze-and-excitation (PSE) module introduces spatial information to each convolutional block, boosting its ability to extract more effective multi-scale features, and the coarse-to-fine (CF) module replaces the conventional decoder to enhance the details of thin vessels and process hard-to-classify pixels again. We evaluated our PC-Net on the Digital Retinal Images for Vessel Extraction (DRIVE) database and an in-house 3D major artery (3MA) database against several recent methods. Our results not only demonstrate the effectiveness of the proposed PSE module and CF module, but also suggest that our proposed PC-Net sets new state of the art in the segmentation of retinal vessels (AUC: 98.31%) and major arteries (AUC: 98.35%) on both databases, respectively.