  We present a sample-variance-limited measurement of the temperature power
spectrum ($TT$) of the cosmic microwave background (CMB) using observations of
a $\sim\! 1500 \,\mathrm{deg}^2$ field made by SPT-3G in 2018. We report
multifrequency power spectrum measurements at 95, 150, and 220GHz covering the
angular multipole range $750 \leq \ell < 3000$. We combine this $TT$
measurement with the published polarization power spectrum measurements from
the 2018 observing season and update their associated covariance matrix to
complete the SPT-3G 2018 $TT/TE/EE$ data set. This is the first analysis to
present cosmological constraints from SPT $TT$, $TE$, and $EE$ power spectrum
measurements jointly. We blind the cosmological results and subject the data
set to a series of consistency tests at the power spectrum and parameter level.
We find excellent agreement between frequencies and spectrum types and our
results are robust to the modeling of astrophysical foregrounds. We report
results for $\Lambda$CDM and a series of extensions, drawing on the following
parameters: the amplitude of the gravitational lensing effect on primary power
spectra $A_\mathrm{L}$, the effective number of neutrino species
$N_{\mathrm{eff}}$, the primordial helium abundance $Y_{\mathrm{P}}$, and the
baryon clumping factor due to primordial magnetic fields $b$. We find that the
SPT-3G 2018 $T/TE/EE$ data are well fit by $\Lambda$CDM with a
probability-to-exceed of $15\%$. For $\Lambda$CDM, we constrain the expansion
rate today to $H_0 = 68.3 \pm 1.5\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and the
combined structure growth parameter to $S_8 = 0.797 \pm 0.042$. The SPT-based
results are effectively independent of Planck, and the cosmological parameter
constraints from either data set are within $<1\,\sigma$ of each other.
(abridged)
