The demands of fault tolerance mean that a wide variety of simple and exotic noise types must be tamed for quantum devices to progress. Real-world hurdles range from imperfect gates and decoherence, to spatiotemporal non-Markovianity, crosstalk, correlated control errors, and undesirable driving of bath transitions. These effects must be treated, but how can we combine everything into a feasible and useful framework, and how can this then be learned? Fortunately, as devices improve, the presence of noise gets sparser and permits more sophisticated models for efficient and general characterisation. Combining recent advances in tensor network learning with non-Markovian process characterisation, we develop efficient techniques to capture a near-universal model for quantum noise. The three fundamental categories we consider are: correlated background dynamics, correlated control imperfections, and control-environment interactions. Our approach to characterise these components employs no assumptions about prior calibrations, and can accommodate large numbers of time-steps and qubits from relatively few experiments. The result is a practical, scalable, and consistent procedure capable of describing arbitrary open dynamics and experimental controls. We stress that the framework is hardware agnostic, but the models are readily modular and can be adapted based on the expected physics of the device or intended purpose of the characterisation. The resulting estimate not only spells out errors, but also serves as a reliable mapping from all experimenter-chosen parameters to circuit outcomes. Hence, this has ready accessibility and applicability to all applied quantum information processing. We bolster these claims with an extensive set of numerical and experimental results. In particular, we demonstrate powerful applications of our method, including bespoke dynamical decoupling sequences, pulse shaping, and optimal logical control choices for any two-qubit gate decomposition. We anticipate this approach to be useful at informing all levels of the quantum control stack: in error suppression, error mitigation, and error correction.