This talk serves as part I of a two-part elementary introduction to mathematical fluid mechanics. In this series of talks we will introduce and (at least partially) derive the Euler and Navier-Stokes equations modeling incompressible fluid flow in two and three dimensions. We will also discuss quantities of interest such as the velocity, pressure, vorticity, and particle trajectory map. For our discussion of inviscid flows modeled by the Euler equations, we will focus in particular on vortex motion, namely Kelvin's circulation theorem and the Helmholtz vortex theorems with applications to movement and stretching of vortex tubes. For viscous flows modeled by the Navier-Stokes equations, we will discuss the no-slip boundary condition, the formation of boundary layers, and convection and diffusion of vorticity. We will mention some interesting theorems as well as some difficult open problems along the way. These talks should be accessible to anyone with a solid background in vector calculus.