Emulating high-accuracy computationally expensive models is crucial for tasks requiring numerous model evaluations, such as uncertainty quantification and optimization. When lower-fidelity models are available, they can be used to improve the predictions of high-fidelity models. Multi-fidelity surrogate models combine information from sources of varying fidelities to construct an efficient surrogate model. However, in real-world applications, uncertainty is present in both high- and low-fidelity models due to measurement or numerical noise, as well as lack of knowledge due to the limited experimental design budget. This paper introduces a comprehensive framework for multi-fidelity surrogate modeling that handles noise-contaminated data and is able to estimate the underlying noise-free high-fidelity model. Our methodology quantitatively incorporates the different types of uncertainty affecting the problem and emphasizes on delivering precise estimates of the uncertainty in its predictions both with respect to the underlying high-fidelity model and unseen noise-contaminated high-fidelity observations, presented through confidence and prediction intervals, respectively. Additionally, the proposed framework offers a natural approach to combining physical experiments and computational models by treating noisy experimental data as high-fidelity sources and white-box computational models as their low-fidelity counterparts. The effectiveness of our methodology is showcased through synthetic examples and a wind turbine application.