Underwater robotic manipulation faces significant challenges due to complex fluid dynamics and unstructured environments, causing most manipulation systems to rely heavily on human teleoperation. In this paper, we introduce AquaBot, a fully autonomous manipulation system that combines behavior cloning from human demonstrations with self-learning optimization to improve beyond human teleoperation performance. With extensive real-world experiments, we demonstrate AquaBot's versatility across diverse manipulation tasks, including object grasping, trash sorting, and rescue retrieval. Our real-world experiments show that AquaBot's self-optimized policy outperforms a human operator by 41% in speed. AquaBot represents a promising step towards autonomous and self-improving underwater manipulation systems. We open-source both hardware and software implementation details.