This paper addresses the generalization issue in deepfake detection by harnessing forgery quality in training data. Generally, the forgery quality of different deepfakes varies: some have easily recognizable forgery clues, while others are highly realistic. Existing works often train detectors on a mix of deepfakes with varying forgery qualities, potentially leading detectors to short-cut the easy-to-spot artifacts from low-quality forgery samples, thereby hurting generalization performance. To tackle this issue, we propose a novel quality-centric framework for generic deepfake detection, which is composed of a Quality Evaluator, a low-quality data enhancement module, and a learning pacing strategy that explicitly incorporates forgery quality into the training process. The framework is inspired by curriculum learning, which is designed to gradually enable the detector to learn more challenging deepfake samples, starting with easier samples and progressing to more realistic ones. We employ both static and dynamic assessments to assess the forgery quality, combining their scores to produce a final rating for each training sample. The rating score guides the selection of deepfake samples for training, with higher-rated samples having a higher probability of being chosen. Furthermore, we propose a novel frequency data augmentation method specifically designed for low-quality forgery samples, which helps to reduce obvious forgery traces and improve their overall realism. Extensive experiments show that our method can be applied in a plug-and-play manner and significantly enhance the generalization performance.