An object pick-and-place system with a camera, a six-degree-of-freedom (DOF) robot manipulator, and a two-finger gripper is implemented based on the robot operating system (ROS) in this paper. A collision-free path planning method is one of the most fundamental problems that has to be solved before the robot manipulator can autonomously pick-and-place objects in complex environments. In the implementation of the real-time pick-and-place system, the success rate and computing time of path planning by a six-DOF robot manipulator are two essential key factors. Therefore, an improved rapidly-exploring random tree (RRT) algorithm, named changing strategy RRT (CS-RRT), is proposed. Based on the method of gradually changing the sampling area based on RRT (CSA-RRT), two mechanisms are used in the proposed CS-RRT to improve the success rate and computing time. The proposed CS-RRT algorithm adopts a sampling-radius limitation mechanism, which enables the random tree to approach the goal area more efficiently each time the environment is explored. It can avoid spending a lot of time looking for valid points when it is close to the goal point, thus reducing the computing time of the improved RRT algorithm. In addition, the CS-RRT algorithm adopts a node counting mechanism, which enables the algorithm to switch to an appropriate sampling method in complex environments. It can avoid the search path being trapped in some constrained areas due to excessive exploration in the direction of the goal point, thus improving the adaptability of the proposed algorithm to various environments and increasing the success rate. Finally, an environment with four object pick-and-place tasks is established, and four simulation results are given to illustrate that the proposed CS-RRT-based collision-free path planning method has the best performance compared with the other two RRT algorithms. A practical experiment is also provided to verify that the robot manipulator can indeed complete the specified four object pick-and-place tasks successfully and effectively.