Cross-correlations between tracers of large-scale structure (LSS), such as galaxies, weak lensing, and thermodynamics of hot gas, provide powerful tests of the cosmological model. In this Ph.D. thesis, we develop analytical models of these tracers and apply them to compare measurements to theoretical predictions of the standard model of cosmology. The complicated non-linear interactions between various components of the Universe present a significant challenge to constraining cosmological or astrophysical models. We aim to maximize the information gained from current and future cosmological datasets in the presence of astrophysical and observational sources of uncertainty. In the first half of the thesis, we describe and validate a hybrid galaxy biasing model (non-linear mapping between dark matter and galaxies) aimed at analyzing the correlations between galaxy positions and weak lensing. We then apply this model to recent data from the Dark Energy Survey, leading to a significant gain in cosmological constraints. In the second half of the thesis, we carry out high significance measurements of cross-correlations between the pressure of hot gas and weak lensing (shear-$y$) and galaxy positions (galaxy-$y$). We constrain the evolution of the average thermal pressure of the Universe and find evidence for reduced pressure in low mass halos. Our results point to the effects of increased baryonic feedback (the impact of supernovae or active galactic nuclei on LSS). These results will help in understanding how baryonic feedback impacts galaxy formation and using the non-linear regime for cosmological analysis with future survey data.