Alzheimer's Disease (AD) is the leading cause of dementia. It results in cortical thickness changes and is associated with a decline in cognition and behaviour. Such decline affects multiple important day-to-day functions, including memory, language, orientation, judgment and problem-solving. Recent research has made important progress in identifying brain regions associated with single outcomes, such as individual AD status and general cognitive decline. The complex projection from multiple brain areas to multiple AD outcomes, however, remains poorly understood. This makes the assessment and especially the prediction of multiple AD outcomes - each of which may unveil an integral yet different aspect of the disease - challenging, particularly when some are not strongly correlated. Here, uniting residual learning, partial least squares (PLS), and predictive modelling, we develop an explainable, generalisable, and reproducible method called the Residual Partial Least Squares Learning (the re-PLS Learning) to (1) chart the pathways between large-scale multivariate brain cortical thickness data (inputs) and multivariate disease and behaviour data (outcomes); (2) simultaneously predict multiple, non-pairwise-correlated outcomes; (3) control for confounding variables (e.g., age and gender) affecting both inputs and outcomes and the pathways in-between; (4) perform longitudinal AD disease status classification and disease severity prediction. We evaluate the performance of the proposed method against a variety of alternatives on data from AD patients, subjects with mild cognitive impairment (MCI), and cognitively normal individuals from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Our results unveil pockets of brain areas in the temporal, frontal, sensorimotor, and cingulate areas whose cortical thickness may be respectively associated with declines in different cognitive and behavioural subdomains in AD. Finally, we characterise re-PLS' geometric interpretation and mathematical support for delivering meaningful neurobiological insights and provide an open software package (re-PLS) available at https://github.com/thanhvd18/rePLS.