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 $1500\text{ }\text{ }{\mathrm{deg}}^{2}$ 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 $300\ensuremath{\le}\ensuremath{\ell}<3000$, 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 $300\ensuremath{\le}\ensuremath{\ell}\ensuremath{\le}1400$ for $EE$ and $300\ensuremath{\le}\ensuremath{\ell}\ensuremath{\le}1700$ for $TE$, resulting in constraints on cosmological parameters comparable to those from other current leading ground-based experiments. We find that the SPT-3G data set is well fit by a $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ cosmological model with parameter constraints consistent with those from Planck and SPTpol data. From SPT-3G data alone, we find ${H}_{0}=68.8\ifmmode\pm\else\textpm\fi{}1.5\text{ }\text{ }\mathrm{km}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ and ${\ensuremath{\sigma}}_{8}=0.789\ifmmode\pm\else\textpm\fi{}0.016$, with a gravitational lensing amplitude consistent with the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ prediction (${A}_{L}=0.98\ifmmode\pm\else\textpm\fi{}0.12$). We combine the SPT-3G and the Planck data sets and obtain joint constraints on the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. The volume of the 68% confidence region in six-dimensional $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ 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.