We present a geometric-analytic introductory treatment of polarization based on the circular polarization basis, which connects directly to the Poincaré sphere. This treatment enables a more intuitive way to arrive at the polarization ellipse from the components of the field. We also present an advanced optics lab that uses Poincaré beams, which have a polarization that is spatially variable. The physics of this lab can reinforce understanding of all states of polarization, and in particular, elliptical polarization. In addition, it exposes students to Laguerre-Gauss modes, the spatial modes used in creating Poincaré beams, which have unique physical properties. In performing this lab, students gain experience in experimental optics, such as aligning and calibrating optical components, using and programming a spatial light modulator, building an interferometer, and performing polarimetry measurements. We present the apparatus for doing the experiments, detailed alignment instructions, and lower-cost alternatives.