Underwater target hunting (UTH) is a critical and complex mission involving the search, monitoring, and hunting of targets in an underwater environment. However, the unpredictable trajectories and flexibility of these targets, along with complex underwater environments, significantly impede the efficiency and success of traditional schemes that depend solely on unmanned underwater vehicles (UUVs). Consequently, this paper presents the “3U network”, a novel heterogeneous framework integrating unmanned aerial vehicles (UAVs), unmanned surface vehicles (USVs), and UUVs for UTH. Within this framework, a UAV searches and monitors the target, a USV acts as a communication relay, and a swarm of UUVs hunts the target. Moreover, to improve the timeliness of target search, we propose the age of information (AoI)-based UAV search strategy. Additionally, we construct a constrained multi-objective optimization problem aiming to minimize energy consumption and mission duration by optimizing vehicles' trajectories, considering mobility limitations, safety, and connectivity constraints. To tackle this problem, we design an AoI- and energy-aware deep reinforcement learning (DRL) algorithm to optimize control policies for heterogeneous vehicles. The experimental results demonstrate that the proposed scheme outperforms the baseline schemes in terms of energy consumption, and mission duration and success rates.