  We study the SU(3) symmetric Fermi-Hubbard model (FHM) in the square lattice
at $1/3$-filling using numerically exact determinant quantum Monte Carlo (DQMC)
and numerical linked-cluster expansion (NLCE) techniques. We present the
different regimes of the model in the $T-U$ plane, which are characterized by
local and short-range correlations, and capture signatures of the
metal-insulator transition and magnetic crossovers. These signatures are
detected as the temperature scales characterizing the rise of the
compressibility, and an interaction-dependent change in the sign of the
diagonal spin-spin correlation function. The analysis of the compressibility
estimates the location of the metal-insulator quantum critical point at $U_c/t
\sim 6$, and provides a temperature scale for observing Mott physics at
finite-$T$. Furthermore, from the analysis of the spin-spin correlation
function we observe that for $U/t \gtrsim6$ and $T \sim J = 4t^2/U$ there is a
development of a short-range two sublattice (2-SL) antiferromagnetic structure,
as well as an emerging three sublattice (3-SL) antiferromagnetic structure as
the temperature is lowered below $T/J \lesssim 0.57$. This crossover from 2-SL
to 3-SL magnetic ordering agrees with Heisenberg limit predictions, and has
observable effects on the density of on-site pairs. Finally, we describe how
the features of the regimes in the $T$-$U$ plane can be explored with
alkaline-earth-like atoms in optical lattices with currently-achieved
experimental techniques and temperatures. The results discussed in this
manuscript provide a starting point for the exploration of the SU(3) FHM upon
doping.
