The robotics community is increasingly interested in autonomous aerial transportation. Unmanned aerial vehicles with suspended payloads have advantages over other systems, including mechanical simplicity and agility, but pose great challenges in planning and control. To realize fully autonomous aerial transportation, this paper presents a systematic solution to address these difficulties. First, we present a real-time planning method that generates smooth trajectories considering the time-varying shape and non-linear dynamics of the system, ensuring whole-body safety and dynamic feasibility. Additionally, an adaptive NMPC with a hierarchical disturbance compensation strategy is designed to overcome unknown external perturbations and inaccurate model parameters. Extensive experiments show that our method is capable of generating high-quality trajectories online, even in highly constrained environments, and tracking aggressive flight trajectories accurately, even under significant uncertainty. We plan to release our code to benefit the community.