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 $300 le ell < 3000$, 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 $300 le ell le 1400$ for $EE$ and $300 le ell le 1700$ for $TE$, resulting in constraints on cosmological parameters comparable to those from other current leading ground-based experiments. We find that the SPT-3G dataset is well-fit by a $Lambda$CDM cosmological model with parameter constraints consistent with those from Planck and SPTpol data. From SPT-3G data alone, we find $H_0 = 68.8 pm 1.5 mathrm{km,s^{-1},Mpc^{-1}}$ and $sigma_8 = 0.789 pm 0.016$, with a gravitational lensing amplitude consistent with the $Lambda$CDM prediction ($A_L = 0.98 pm 0.12$). We combine the SPT-3G and the Planck datasets and obtain joint constraints on the $Lambda$CDM model. The volume of the 68% confidence region in six-dimensional $Lambda$CDM parameter space is reduced by a factor of 1.5 compared to Planck-only constraints, with only slight 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.
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