Cosmological correlators hold the key to high-energy physics as they probe the earliest moments of our Universe, and conceal hidden mathematical structures. However, even at tree-level, perturbative calculations are limited by technical difficulties absent in flatspace Feynman diagrammatics. In this paper, we introduce CosmoFlow: a new accurate open source Python code that computes tree-level cosmological correlators by tracing their time flow. This code is specifically designed to offer a simple, intuitive and flexible coding environment to theorists, primordial and late-time cosmologists. It can typically serve to complement analytical computations, to provide physical intuition when studying various inflationary theories, and to obtain exact results in regimes that are analytically out of reach. This paper presents the basic structure of CosmoFlow, leads the reader through an in-depth user-guide, and illustrates how it can be used with a series of worked examples. Our hope is that this first building block sets the stage for a bank of theoretical data, which can be nurtured and enhanced collaboratively by the community. CosmoFlow is publicly available on GitHub.