Autonomous drone navigation in dynamic environments remains a critical challenge, especially when dealing with unpredictable scenarios including fast-moving objects with rapidly changing goal positions. While traditional planners and classical optimisation methods have been extensively used to address this dynamic problem, they often face real-time, unpredictable changes that ultimately leads to sub-optimal performance in terms of adaptiveness and real-time decision making. In this work, we propose a novel motion planner, AgilePilot, based on Deep Reinforcement Learning (DRL) that is trained in dynamic conditions, coupled with real-time Computer Vision (CV) for object detections during flight. The training-to-deployment framework bridges the Sim2Real gap, leveraging sophisticated reward structures that promotes both safety and agility depending upon environment conditions. The system can rapidly adapt to changing environments, while achieving a maximum speed of 3.0 m/s in real-world scenarios. In comparison, our approach outperforms classical algorithms such as Artificial Potential Field (APF) based motion planner by 3 times, both in performance and tracking accuracy of dynamic targets by using velocity predictions while exhibiting 90% success rate in 75 conducted experiments. This work highlights the effectiveness of DRL in tackling real-time dynamic navigation challenges, offering intelligent safety and agility.