This course is the first, introductory part of our two week course proposal for ESSLLI 2015. The course introduces a number of logics that are important in many modern scientific disciplines: propositional logic, modal and temporal logics, description logics, dynamic logics, firstorder logic, fixedpoint logic and higherorder logic. We will introduce them in a way that makes clear both the underlying unity of these systems, their differences, possible uses and applications. Beginning with propositional logic, we present logical languages of increasing strength for talking about relational structures, and in parallel we study inference tasks (satisfiability, model checking, model building, etc.) and algorithms for them. The instructors believe that the course material is indispensable to a modern education in logic. Unfortunately, much of it is only currently available in advanced courses and textbooks. This course is designed to make this material accessible to an interdisciplinary audience, and it will be based on a textbook that is currently being written by the two instructors. We intend to make a draft version of our text available to course participants.