Here's the general rule I use in my own calculations.
In cartesian like coordinates $x^{\mu} \sim \mathrm{L}^1$ :
\begin{align}
g_{\mu \nu} &\sim \mathrm{L}^0, \\[12pt]
\Gamma_{\mu \nu}^{\lambda} &\sim \mathrm{L}^{-1}, \\[12pt]
R^{\lambda}_{\; \kappa \mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt]
R_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt]
G_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt]
R &\sim \mathrm{L}^{-2}, \\[12pt]
T_{\mu \nu} &\sim \mathrm{L}^{-4}, \\[12pt]
\kappa \equiv 8 \pi G &\sim \mathrm{L}^{2}, \\[12pt]
\end{align}
Take note that the dimensions of these quantities heavily depend on the dimensions of your coordinates, which are totally arbitrary. However, whatever the coordinates $x^{\mu}$, you have
\begin{equation}
ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu} \sim \mathrm{L}^2.
\end{equation}
Also, tensor invariants do not depend on the coordinates and have the same dimensions in all coordinates systems. For example:
\begin{align}
R \equiv g^{\mu \nu} \, R_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt]
R^{\mu \nu \lambda \kappa} \, R_{\mu \nu \lambda \kappa} &\sim \mathrm{L}^{-4}.
\end{align}