This paper presents an optimal operational control strategy for a multi-energy system. First, the economic performance of the multi-energy system and the way it participates in different energy markets is investigated. Then, a two-stage robust stochastic optimization model is proposed with the objective of minimizing the total system operating cost and compensating the operational risk associated with the existing uncertainty considering the grid operating constraints. Then, to satisfy the dynamic power balance constraint of the day-ahead market, an interaction management mechanism is used to deal with the power imbalance in the real-time market. Next, ancillary services are provided according to the optimization model, i.e., market-based demand response planning and direct load control services. In addition, multiple energy network synergies are incorporated into the system flexibility strategy decisions. Finally, the feasibility of the proposed strategy is verified by a set of case studies. The simulation results show that the total operating cost of the multi-energy system is reduced by at least 9% considering multiple economic coordination, validating the effectiveness of the proposed approach.