The emerging multi-energy system has brought new challenges and opportunities to energy business worldwide. To address the issues in multi-energy trading, this paper proposes an electricity, heating, and cooling trading model for the interaction between the multi-energy service provider (MESP) and multi-energy consumer (MEC) by using bi-level programming. In the upper level, the model instructs the MESP to make decisions on the optimal energy purchasing scheme and energy economic dispatch. In the lower level, optimal consuming patterns of different energies with the given retail prices are modeled for the MEC. Specifically, a novel multi-energy demand response (DR) program that employs energy conversion devices is proposed. Numerical results show that MEC can reduce its consumption cost via the multi-energy DR. Meanwhile, MESP benefits greatly from the flexibilities of energy conversion. This research can provide theoretical support for the future development of multi-energy trading.