  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.
