Robotic caregiving is emerging as a critical application domain where human-centered robotics, learning-based manipulation, and embodied intelligence converge. As global aging accelerates, assistive robots are expected to support essential daily activities, such as feeding, grooming, dressing, and body repositioning, which demand precise, adaptive, and safe physical interaction with users. Despite rapid progress, these tasks remain challenging due to deformable objects, diverse human behaviors, and safety-critical, contact-rich dynamics. This survey provides the first unified review of learning-based robotic caregiving across all four core manipulation tasks. We analyze recent advances through a cross-cutting framework grounded in four major learning paradigms: multimodal perception, imitation learning, reinforcement learning, and embodied reasoning. For each caregiving task, we synthesize how these paradigms address unique perceptual, physical, and safety requirements, identifying shared principles and task-specific characteristics. We also examine simulation platforms as enablers for scalable training and reproducible evaluation. This survey further summarizes a conceptual unified perception-reasoning-control loop that distills common structures across caregiving tasks. Finally, we highlight open challenges, including user-centered personalization, safety validation, and ethical deployment, and outline research directions toward trustworthy and adaptive assistive robots.