We present measurements of the E-mode (EE) polarization power spectrum and temperature-E-mode (TE) cross-power spectrum of the cosmic microwave background using data collected by SPT-3G, the latest instrument installed on the South Pole Telescope. This analysis uses observations of a 1500 deg 2 region at 95, 150, and 220 GHz taken over a four-month period in 2018. We report binned values of the EE and TE power spectra over the angular multipole range 300 ≤ l < 3000, using the multifrequency data to construct six semi-independent estimates of each power spectrum and their minimum-variance combination. These measurements improve upon the previous results of SPTpol across the multipole ranges 300 ≤ l ≤ 1400 for EE and 300 ≤ l ≤ 1700 for TE, resulting in constraints on cosmological parameters comparable to those from other current leading ground-based experiments. We find that the SPT-3G data set is well fit by a ΛCDM cosmological model with parameter constraints consistent with those from Planck and SPTpol data. From SPT-3G data alone, we find H 0 ¼ 68.8 AE 1.5 km s −1 Mpc −1 and σ 8 ¼ 0.789 AE 0.016, with a gravitational lensing amplitude consistent with the ΛCDM prediction (A L ¼ 0.98 AE 0.12). We combine the SPT-3G and the Planck data sets and obtain joint constraints on the ΛCDM model. The volume of the 68% confidence region in six-dimensional ΛCDM parameter space is reduced by a factor of 1.5 compared to Planck-only constraints, with no significant shifts in central values. We note that the results presented here are obtained from data collected during just half of a typical observing season with only part of the focal plane operable, and that the active detector count has since nearly doubled for observations made with SPT-3G after 2018.