Building an emulator for multi-fidelity simulations is becoming increasingly important in science. A novel multi-level procedure is introduced for effectively combining cheaply-obtained, but biased and noisy simulations, with expensive but less biased simulations, which builds an accurate emulator in multiple steps for the highest-fidelity simulations. Theoretical properties, such as the nominal error, are rigorously studied, which shows theoretically that substantial gains in emulation accuracy can be attained with the multi-level approach. Based on the theoretical results, an experimental plan for multi-fidelity simulations which accounts for each simulation cost is also developed. The performance is examined by numerical studies, which demonstrate that the emulator saves substantial computational cost while retaining emulation accuracy for the highest-fidelity simulations.