The ever-burgeoning growth of autonomous unmanned aerial vehicles (UAVs) has demonstrated a promising platform for utilization in real-world applications. In particular, a UAV equipped with a vision system could be leveraged for surveillance applications. This paper proposes a learning-based UAV system for achieving autonomous surveillance, in which the UAV can be of assistance in autonomously detecting, tracking, and following a target object without human intervention. Specifically, we adopted the YOLOv4-Tiny algorithm for semantic object detection and then consolidated it with a 3D object pose estimation method and Kalman filter to enhance the perception performance. In addition, UAV path planning for a surveillance maneuver is integrated to complete the fully autonomous system. The perception module is assessed on a quadrotor UAV, while the whole system is validated through flight experiments. The experiment results verified the robustness, effectiveness, and reliability of the autonomous object tracking UAV system in performing surveillance tasks. The source code is released to the research community for future reference.