With the increasing penetration of renewable energy resources in microgrids, intelligent energy management has become increasingly important due to the intermittent nature of solar and wind generation. This study presents a deep reinforcement learning-based energy management framework for a grid-connected microgrid consisting of photovoltaic panels, a wind turbine, a battery energy storage system, a diesel generator, and the utility grid. The main novelty of this work lies in combining Double Deep Q-Network (Double DQN) with Prioritized Experience Replay (PER) and designing a multi-objective reward function that simultaneously considers operating cost, carbon emissions, supply reliability, and battery state-of-charge management before peak-price hours. This structure enables the agent to learn not only an economic operating policy but also decisions that adapt to the temporal patterns of renewable generation and electricity prices. The proposed model is evaluated over a 24-hour scheduling horizon after 2,000 training episodes. The results show satisfactory convergence, reducing the daily operating cost from approximately $10 to $3.44, corresponding to a 66% cost reduction. In addition, total carbon emissions decrease from nearly 41 kg CO₂ to 16.6 kg CO₂, representing about a 60% reduction. Moreover, load shedding is completely eliminated, and the demand is fully supplied throughout the day. Battery behavior analysis indicates that the agent charges the battery during periods of high renewable generation and discharges it during peak-price hours. The results confirm the effectiveness and scalability of the proposed method for energy management of renewable-rich microgrids.