The angular power spectra of the three types of two-point correlation functions obtained from the cosmic microwave background (CMB) temperature and E-mode polarization anisotropies (the TT/TE/EE spectra) encode information on the composition and evolution of the universe. Measurements of the spectra contributed to establishing the LCDM cosmological model as the standard, and the satellite experiment Planck achieved subpercent-precision constraints on parameters of LCDM by mainly using its TT/TE spectra at large angular scales. However, there are also challenges to LCDM, a major one being the Hubble tension. Better measurements of the TE/EE spectra at small angular scales by other experiments like the South Pole Telescope (SPT) can serve as a powerful consistency test of Planck’s results and provide new insights on the tension. Since 2019, we have been taking data consistently with SPT-3G, the third-generation imaging instrument installed on the telescope in 2017. My thesis project is about analyzing data from 2019 and 2020 to substantially improve SPT-3G's existing measurements of the TE/EE spectra and constraints on cosmological parameters based on data from 2018 and to prepare for a future analysis based on the full dataset from SPT-3G. The analysis comprises three major steps: making CMB anisotropy maps from time series recorded by detectors, calculating the TT/TE/EE spectra from the maps, and fitting cosmological models to the spectra. As of the writing of this thesis in February 2024, we have produced maps and spectra and are working on constraining cosmological models. Uncertainties in the TE/EE spectra obtained from this new dataset are smaller than those from the 2018 dataset by an order of magnitude at small angular scales, and we are excited to significantly improve our existing constraints on cosmological parameters such as the Hubble constant. This thesis comprises four chapters. In chapter 1, I will discuss the origin of the three types of spectra, current measurements of the spectra and constraints on LCDM parameters, and future directions in the field. After that, I will introduce SPT-3G in chapter 2 by discussing important properties of the instrument and showing example raw data. Then, in chapter 3, I will discuss the dataset used in the analysis and methods involved in each major step of the analysis, In the final chapter, chapter 4, I will discuss further details of some of the methods and report results from them, show our maps and spectra, and discuss tasks that remain to be done.