This study proposes a multi-objective optimization framework for energy management in a grid-connected microgrid comprising photovoltaic (PV) panels, wind turbines (WT), diesel generators (DG), battery energy storage systems (BESS), fuel cells (FC), and utility-grid interaction. The proposed framework incorporates a source-prioritized Home-to-Grid (H2G) strategy with an intelligent source-selection algorithm, enabling surplus renewable energy to be exported to the utility grid while maximizing economic benefits and maintaining secure system operation. The primary objectives are to minimize total operating cost and carbon dioxide (CO2) emissions while ensuring reliable and continuous power supply. To achieve these objectives, a two-layer optimization architecture is developed, where the global layer employs Mixed-Integer Linear Programming (MILP) for optimal day-ahead energy scheduling, and the local layer coordinates renewable generation and battery operation to improve operational feasibility and resource utilization. The conflicting objectives are handled using the Tchebycheff scalarization method, which provides normalized objective balancing, reduces scale dependency, and enables a more comprehensive exploration of the Pareto-optimal solution space. The proposed framework is evaluated over a 24-hour scheduling horizon under three representative operational configurations. The complete hybrid configuration (PV-WT-FC-DG-BESS-Grid) achieves a Pareto frontier spanning approximately 600–5200 kg CO2 while reducing the operating cost from 2180 USD to 760 USD. For a representative operating point, the proposed framework achieves an operating cost of 1396.62 USD with CO2 emissions of 1897.40 kg, outperforming reduced system configurations. Comparative evaluation further demonstrates that the proposed Tchebycheff scalarization approach provides improved Pareto-front distribution, enhanced preservation of extreme trade-off solutions, reduced sensitivity to objective scaling, and greater decision-making flexibility than the conventional Weighted Sum and ϵ[jls-end-space/]-Constraint approaches. Moreover, extended 30-day simulations confirm stable scheduling behavior while satisfying all operational constraints and reducing dependence on diesel generation and utility-grid imports. Overall, the proposed framework provides an effective and reliable decision-support tool for economically efficient and environmentally sustainable operation of renewable-rich grid-connected microgrids.