UAV swarm systems exhibit distinctive advantages including efficient coordination, robust survivability, and wide-area coverage capabilities. Path-following and formation control constitute critical technologies essential for achieving multi-UAV cooperative missions, playing a pivotal role in enhancing mission performance and flight safety. This paper addresses path-following guidance, formation construction, maintenance, and obstacle avoidance for UAV swarms in three-dimensional space. A cooperative control method based on hybrid gradient vector fields is proposed, consisting of a path-following algorithm for the leader UAV and a formation control strategy for the follower UAVs. The method plans a guidance path with an optimal entry angle for the leader UAV, enabling it to efficiently approach the mission area and follow three-dimensional paths. The follower UAVs dynamically adjust their attitudes and airspeeds based on the relative positions of other UAVs, maintaining the predetermined formation structure. Obstacle avoidance is implemented via the three-dimensional vector field histogram algorithm, ensuring the safety of swarm flight. Simulations demonstrate that the proposed method exhibits good adaptability and robustness, providing an effective solution for the cooperative control of UAV swarms.