The low carbon transition requires the high growth of renewable generation penetration in energy systems to ultimately achieve net-zero carbon target. To ensure the reliable operation of energy systems with high intermittent renewable output, it is critical to have sufficient flexible resources to avoid curtailment. Therefore, the integrated power-natural gas-heating energy systems with power to gas (P2G) and gas storage has attracted great research interest especially on the combined operation method to enhance the flexibility provision between each other. In this paper, taking heating demand, P2G and gas storage into consideration, a multi-objective optimal operation strategy of integrated power-natural gas-heating energy systems is presented to obtain the maximum economic and environmental benefits. Furthermore, a novel model of flexibility metric is proposed based on redundant linepack and gas storage. Case studies without P2G and with P2G are carried out on integrated IEEE 39-bus power and Belgian 20-node gas system. Simulation results demonstrate that P2G not only can be beneficial for operation of the integrated energy systems in terms of total operational cost decline from M$2.510 to M$2.503, CO2 emission reduction from 62,860 ton to 62,240 ton and wind curtailment decrease from 25.58% to 4.22% but also has significant effect on flexibility improvement of a 71.72% increase.