This paper presents a novel autonomous drone-based smoke plume tracking system capable of navigating and tracking plumes in highly unsteady atmospheric conditions. The system integrates advanced hardware and software and a comprehensive simulation environment to ensure robust performance in controlled and real-world settings. The quadrotor, equipped with a high-resolution imaging system and an advanced onboard computing unit, performs precise maneuvers while accurately detecting and tracking dynamic smoke plumes under fluctuating conditions. Our software implements a two-phase flight operation: descending into the smoke plume upon detection and continuously monitoring the smoke's movement during in-plume tracking. Leveraging Proportional Integral-Derivative (PID) control and a Proximal Policy Optimization (PPO) based Deep Reinforcement Learning (DRL) controller enables adaptation to plume dynamics. Unreal Engine simulation evaluates performance under various smoke-wind scenarios, from steady flow to complex, unsteady fluctuations, showing that while the PID controller performs adequately in simpler scenarios, the DRL-based controller excels in more challenging environments. Field tests corroborate these findings. This system opens new possibilities for drone-based monitoring in areas like wildfire management and air quality assessment. The successful integration of DRL for real-time decision-making advances autonomous drone control for dynamic environments.