With the rapid growth of energy demand and the increasing severity of environmental pollution, low-carbon and clean energy have become the mainstream direction of energy development. An integrated energy system (IES) is one of the effective ways to reduce environmental pollution and improve energy efficiency. Based on this, this paper proposes a coupled model that considers two-stage power-to-gas (P2G) and carbon capture system (CCS) and extends it to IES, including electricity, heat, and gas. Then, a low-carbon optimization scheduling model for IES based on the Stackelberg game is established. In this model, the IES operator is regarded as the leader, and the user coalition is regarded as the follower. The leader optimizes the energy selling price and equipment outputs with the goal of maximizing profit, while the follower optimizes the energy consumption strategy with the goal of minimizing energy purchased cost. The genetic algorithm and CPLEX solver are combined to solve the model, and eight scenarios are analyzed. The results indicate that the model proposed can fully tap the potential of demand response. In addition, the introduction of two-stage P2G makes the wind power absorption rate reach 99.31%, and the CO2 captured by CCS could be used to synthesize natural gas, reducing carbon emission.