As the proportion of new energy sources, such as wind power, in the electricity system rapidly increases, their participation in spot market competition has become an inevitable trend. However, the uncertainty of clearing price and wind power output will lead to bidding deviation and bring revenue risks. In response to this, a bidding strategy is proposed for wind farms to participate in the spot market jointly with carbon capture power plants (CCPP) that have flexible regulation capabilities. First, a two-stage decision model is constructed in the day-ahead market and real-time balancing market. Under the joint bidding mode, CCPP can help alleviate wind power output deviations, thereby reducing real-time imbalanced power settlement. On this basis, a tiered carbon trading mechanism is introduced to optimize day-ahead bidding, aiming at maximizing revenue in both the electricity spot market and carbon trading market. Secondly, conditional value at risk (CVaR) is introduced to quantitatively assess the risks posed by uncertainties in the two-stage decision model, and the risk aversion coefficient is used to represent the decision-maker’s risk preference, providing corresponding strategies. The model is transformed into a mixed-integer linear programming model using piecewise linearization and McCormick enveloping. Finally, the effectiveness of the proposed model and methods is verified through numerical examples.