  We present a dedicated transit and radial velocity survey of planets orbiting
subgiant stars observed by the TESS Mission. Using $\sim$$16$ nights on
Keck/HIRES, we confirm and characterize $12$ new transiting planets -- $\rm
TOI-329\,b$, $\rm HD\,39688\,b$ ($\rm TOI-480$), $\rm TOI-603\,b$, $\rm
TOI-1199\,b$, $\rm TOI-1294\,b$, $\rm TOI-1439\,b$, $\rm TOI-1605\,b$, $\rm
TOI-1828\,b$, $\rm HD\,148193\,b$ ($\rm TOI-1836$), $\rm TOI-1885\,b$, $\rm
HD\,83342\,b$ ($\rm TOI-1898$), $\rm TOI-2019\,b$ -- and provide updated
properties for 9 previously confirmed TESS subgiant systems ($\rm TOI-197$,
$\rm TOI-954$, $\rm TOI-1181$, $\rm TOI-1296$, $\rm TOI-1298$, $\rm TOI-1601$,
$\rm TOI-1736$, $\rm TOI-1842$, $\rm TOI-2145$). We also report the discovery
of an outer, non-transiting planet, $\rm TOI-1294\,c$ ($P=160.1\pm2.5$ days,
$M_{\mathrm{p}}=148.3^{+18.2}_{-16.4} \,M_{\oplus}$), and three additional
stars with long-term RV trends. We find that at least $19\pm8\%$ of subgiants
in our sample of $21$ stars have outer companions, comparable to main-sequence
stars. We perform a homogeneous analysis of the stars and planets in the
sample, with median uncertainties of $3\%$, $8\%$ and $15\%$ for planet radii,
masses and ages, doubling the number of known planets orbiting subgiant stars
with bulk densities measured to better than $10\%$. We observe a dearth of
giant planets around evolved stars with short orbital periods, consistent with
tidal dissipation theories that predict the rapid inspiral of planets as their
host stars leave the main sequence. We note the possible evidence for two
distinct classes of hot Jupiter populations, indicating multiple formation
channels to explain the observed distributions around evolved stars. Finally,
continued RV monitoring of planets in this sample will provide a more
comprehensive understanding of demographics for evolved planetary systems.
