A comprehensive understanding of the behaviours of the various geophysical processes and an effective evaluation of time series (else referred to as "stochastic") simulation models require, among others, detailed investigations across temporal scales. In this work, we propose a novel and detailed methodological framework for advancing and enriching such investigations in a hydroclimatic context. This specific framework is primarily based on a new feature compilation for multi-scale hydroclimatic analyses, and can facilitate largely interpretable feature investigations and comparisons in terms of temporal dependence, temporal variation, "forecastability", lumpiness, stability, nonlinearity (and linearity), trends, spikiness, curvature and seasonality. Multifaceted characterizations are herein obtained by computing the values of the proposed feature compilation across nine temporal resolutions (i.e., the 1-day, 2-day, 3-day, 7-day, 0.5-month, 1-month, 2-month, 3-month and 6-month ones) and three hydroclimatic time series types (i.e., temperature, precipitation and streamflow) for 34-year-long time series records originating from 511 geographical locations across the contiguous United States. Based on the acquired information and knowledge, similarities and differences between the examined time series types with respect to the evolution patterns characterizing their feature values with increasing (or decreasing) temporal resolution are identified. Moreover, the computed features are used as inputs to unsupervised random forests for detecting any meaningful clusters between the examined hydroclimatic time series. This clustering plays an illustrative role within this research, as it facilitates the identification of spatial patterns (with them consisting an important scientific target in hydroclimatic research) and their cross-scale comparison...