Operator learning for complex nonlinear systems is increasingly common in modeling multi-physics and multi-scale systems such as climate modeling. However, training such high-dimensional operators requires a large amount of expensive, high-fidelity data, either from experiments or simulations. In many cases, we may not have access to sufficient high-fidelity data to train, however we may have a large amount of low-fidelity that that is less accurate with greater uncertainty associated with it. The question is how to combine the low-fidelity and high-fidelity data to create a model that is capable of training more accurately than using the low- or high-fidelity data alone. In this work, we present a composite Deep Operator Network (DeepONet) for learning using two datasets with different levels of fidelity to accurately learn complex operators when sufficient high-fidelity data is not available. Additionally, we demonstrate that the presence of low-fidelity data can improve the predictions of physics-informed learning with DeepONets. We demonstrate the new multi-fidelity training in diverse examples, including modeling of the ice-sheet dynamics of the Humboldt glacier, Greenland, using two different fidelity models and also using the same physical model at two different resolutions. We will discuss extensions of the multifidelity framework, such as how multifidelity learning can contribute to more accurate training even in the absence of data, with only physics used to train.