Manipulator kinematics is concerned with the motion of each link within a manipulator without considering mass or force. In this article, which is the first in a two-part tutorial, we provide an introduction to modelling manipulator kinematics using the elementary transform sequence (ETS). Then we formulate the first-order differential kinematics, which leads to the manipulator Jacobian, which is the basis for velocity control and inverse kinematics. We describe essential classical techniques which rely on the manipulator Jacobian before exhibiting some contemporary applications. Part II of this tutorial provides a formulation of second and higher-order differential kinematics, introduces the manipulator Hessian, and illustrates advanced techniques, some of which improve the performance of techniques demonstrated in Part I. We have provided Jupyter Notebooks to accompany each section within this tutorial. The Notebooks are written in Python code and use the Robotics Toolbox for Python, and the Swift Simulator to provide examples and implementations of algorithms. While not absolutely essential, for the most engaging and informative experience, we recommend working through the Jupyter Notebooks while reading this article. The Notebooks and setup instructions can be accessed at https://github.com/jhavl/dkt.