A novel framework is proposed for the trajectory design of multiple unmanned aerial vehicles (UAVs) based on the prediction of users' mobility information. The problem of joint trajectory design and power control is formulated for maximizing the instantaneous sum transmit rate while satisfying the rate requirement of users. In an effort to solve this pertinent problem, a three-step approach is proposed which is based on machine learning techniques to obtain both the position information of users and the trajectory design of UAVs. Firstly, a multi-agent Qlearning based placement algorithm is proposed for determining the optimal positions of the UAVs based on the initial location of the users. Secondly, in an effort to determine the mobility information of users based on a real dataset, their position data is collected from Twitter to describe the anonymous usertrajectories in the physical world. In the meantime, an echo state network (ESN) based prediction algorithm is proposed for predicting the future positions of users based on the real dataset. Thirdly, a multi-agent Q-learning based algorithm is conceived for predicting the position of UAVs in each time slot based on the movement of users. In this algorithm, multiple UAVs act as agents to find optimal actions by interacting with their environment and learn from their mistakes. Additionally, we also prove that the proposed multi-agent Q-learning based trajectory design and power control algorithm can converge under mild conditions. Numerical results are provided to demonstrate that as the size of the reservoir increases, the proposed ESN approach improves the prediction accuracy. Finally, we demonstrate that throughput gains of about 17% are achieved.