This book serves as a companion reading for the first year of mathematics studies and beyond. It focuses on the motivation and explanation of central concepts through examples and exemplary results. Starting from the most elementary and historically earliest mathematical concepts of measurement and counting, you will be guided to modern concepts such as metrics, measures, and vector spaces, which are then applied to solve concrete problems. The selection of material is designed to highlight the connections between different introductory lectures, which are often overlooked in regular study programs. The book is aimed at readers who are familiar with elementary set theory as a language for describing mathematical content, as well as with real numbers as a mathematical concept. This corresponds to a certain level of knowledge.