Growing concerns about global greenhouse gas emissions have led power systems to utilize clean and highly efficient resources. In the meantime, renewable energy plays a vital role in energy prospects worldwide. However, the random nature of these resources has increased the demand for energy storage systems. On the other hand, due to the higher efficiency of multi-energy systems compared to single-energy systems, the development of such systems, which are based on different types of energy carriers, will be more attractive for the utilities. Thus, this paper represents a multi-objective assessment for the operation of a multi-carrier microgrid (MCMG) in the presence of high-efficiency technologies comprising compressed air energy storage (CAES) and power-to-gas (P2G) systems. The objective of the model is to minimize the operation cost and environmental pollution. CAES has a simple-cycle mode operation besides the charging and discharging modes to provide more flexibility in the system. Furthermore, the demand response program is employed in the model to mitigate the peaks. The proposed system participates in both electricity and gas markets to supply the energy requirements. The weighted sum approach and fuzzy-based decision-making are employed to compromise the optimum solutions for conflicting objective functions. The multi-objective model is examined on a sample system, and the results for different cases are discussed. The results show that coupling CAES and P2G systems mitigate the wind power curtailment and minimize the cost and pollution up to 14.2% and 9.6%, respectively.