  We report the magnetic, magnetoelastic, and magnetotransport properties of
single crystals of polar magnets $R$AuGe ($R=$ Dy, Ho, and Gd), grown by Au-Ge
self-flux. Magnetization and magnetostriction measurements reveal multi-step
metamagnetic transitions for the $c$-axis magnetic field ($H\parallel c$) for
DyAuGe and HoAuGe, suggesting magnetic frustration in the triangular lattice of
$R$ ions. The magnetic phase diagrams have clarified a close connection between
the magnetoelastic property and the emergence of the intermediate metamagentic
phase. The magnetic-field dependence of the resistivity and Hall resistivity
reveal the semimetallic transport dominated by hole-type carriers, consistent
with the behavior in a nonmagnetic analogue YAuGe. We also identify a signature
of an anomalous Hall effect (AHE) proportional to the field-induced
magnetization in $R=$ Dy, Ho, and Gd. GdAuGe shows magnetic and transport
behavior as reported in a previous study using Bi-flux grown single crystals,
while the self-flux grown crystal shows larger magnetoresistance ($\sim$ 345\%,
at 1.8 K and 9 T) due to higher hole-type carrier mobility ($\sim$ 6400
cm$^2$/Vs). Using the two-band model analysis considering the mobility change
during the magnetization process, we extract the anomalous Hall conductivity:
$\sim 1200$ S/cm and $\sim 530$ S/cm for $R=$ Dy and Ho, respectively, at 1.8 K
with 9 T for $H\parallel c$. The magnitude of conductivity suggests a
contribution of intrinsic origin, possibly related to the Berry curvature in
the electron bands induced by the time-reversal symmetry breaking and the polar
lattice.
