Single crystals of $R$Mg$_{2}$Cu$_{9}$ ($R$=Y, Ce-Nd, Gd-Dy, Yb) were grown using a high-temperature solution growth technique and were characterized by measurements of room-temperature x-ray diffraction, temperature-dependent specific heat and temperature-, field-dependent resistivity and anisotropic magnetization. YMg$_{2}$Cu$_{9}$ is a non-local-moment-bearing metal with an electronic specific heat coefficient, $gamma sim$ 15 mJ/mol K$^2$. Yb is divalent and basically non-moment bearing in YbMg$_{2}$Cu$_{9}$. Ce is trivalent in CeMg$_{2}$Cu$_{9}$ with two magnetic transitions being observed at 2.1 K and 1.5 K. PrMg$_{2}$Cu$_{9}$ does not exhibit any magnetic phase transition down to 0.5 K. The other members being studied ($R$=Nd, Gd-Dy) all exhibits antiferromagnetic transitions at low-temperatures ranging from 3.2 K for NdMg$_{2}$Cu$_{9}$ to 11.9 K for TbMg$_{2}$Cu$_{9}$. Whereas GdMg$_{2}$Cu$_{9}$ is isotropic in its paramagnetic state due to zero angular momentum ($L$=0), all the other local-moment-bearing members manifest an anisotropic, planar magnetization in their paramagnetic states. To further study this planar anisotropy, detailed angular-dependent magnetization was carried out on magnetically diluted (Y$_{0.99}$Tb$_{0.01}$)Mg$_{2}$Cu$_{9}$ and (Y$_{0.99}$Dy$_{0.01}$)Mg$_{2}$Cu$_{9}$. Despite the strong, planar magnetization anisotropy, the in-plane magnetic anisotropy is weak and field-dependent. A set of crystal electric field parameters are proposed to explain the observed magnetic anisotropy.
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