The redshifted 21 cm line promises to provide a wealth of information about the evolution of our universe but remains as yet undetected. The general theme of this thesis is developing increasingly realistic models of the raw data we collect from a radio telescope. This is important because at the end of the day extracting the cosmological signal from the data will be accomplished by achieving a level of understanding of all the possible alternative sources that might mimic the cosmological signal, to a degree that we can confidently reject those alternatives as causes of our detection. The work presented in this thesis has been done in the context of working on the HERA experiment which aims to make the first measurements of spatial fluctuations in the emission from neutral hydrogen. In this thesis I emphasized aspects of the visibility function that are important for efficient and realistic visibility simulations including full account of polarization effects, in particular using a harmonic parameterization of the integrand. I assessed the effect of potential ionospheric attenuation on the suppression of polarization contamination in 21 cm power spectrum measurements using visibility simulations based on historical ionospheric plasma density data. I showed how we can use closed-form calculations of the cross-frequency angular power spectrum on the sky to generate simple mock cosmological signal simulations that are useful for validating data analysis methods. I showed how the window functions associated with a 21 cm power spectrum estimate can be approximated by simple forms that are much cheaper to evaluate than the general definition. Finally, I produced a new Southern Sky Model that combines the best available diffuse radio emission surveys that cover HERA's field of view and observing bandwidth with a point source catalog without double counting flux.