  We present the results of a thorough investigation of the physical properties
of EuAg$_4$Sb$_2$ single crystals using magnetization, heat capacity, and
electrical resistivity measurements. High-quality single crystals, which
crystallize in a trigonal structure with space group $R\bar{3}m$, were grown
using a conventional flux method. Temperature-dependent magnetization
measurements along different crystallographic orientations confirm two
antiferromagnetic phase transitions around $T_{N1}$ = 10.5 K and $T_{N2}$ = 7.5
K. Isothermal magnetization data exhibit several metamagnetic transitions below
these transition temperatures. Antiferromagnetic phase transitions in
EuAg$_4$Sb$_2$ are further confirmed by two sharp peaks in the
temperature-dependent heat capacity data at $T_{N1}$ and $T_{N2}$, which shift
to the lower temperature in the presence of an external magnetic field. Our
systematic heat capacity measurements utilizing a long-pulse and single-slope
analysis technique allow us to detect a first-order phase transition in
EuAg$_4$Sb$_2$ at 7.5 K. The temperature-dependent electrical resistivity data
also manifest two features associated with magnetic order. The
magnetoresistance exhibits a broad hump due to the field-induced metamagnetic
transition. Remarkably, the magnetoresistance keeps increasing without showing
any tendency to saturate as the applied magnetic field increases, and it
reaches $\sim$20000\% at 1.6 K and 60 T. At high magnetic fields, several
magnetic quantum oscillations are observed, indicating a complex Fermi surface.
A large negative magnetoresistance of about -55\% is also observed near
$T_{N1}$. Moreover, the $H$-$T$ phase diagram constructed using magnetization,
heat capacity, and magnetotransport data indicates complex magnetic behavior in
EuAg$_4$Sb$_2$.
