  The intriguing interplay between the two $e_g$ orbitals in the high-$T_c$
nickelate superconductor La$_3$Ni$_2$O$_7$ is studied via the density matrix
renormalization group (DMRG) and finite-$T$ thermal tensor-network
calculations. We consider a bilayer $t$-$J$ model that incorporates both
$d_{x^2-y^2}$ and $d_{z^2}$ orbitals, as well as inter-orbital hybridization
and ferromagnetic Hund's rule coupling. Our DMRG calculations reveal that both
orbitals exhibit algebraic superconductive (SC) pairing correlations in the
ground state, with the former being stronger than the latter. To confirm the
dominant $d_{x^2-y^2}$-orbital SC order, we compute the pairing susceptibility
$\chi_{\rm SC}$, and find a power-law divergence of $\chi_{\rm SC}$ below
$T_c^* \simeq 0.03 t_c$, with $t_c$ being the intralayer hopping of the
$d_{x^2-y^2}$ orbital. Regarding the $d_{z^2}$ orbitals, their finite-$T$
$\chi_{\rm SC}$ results show weaker divergence and suggest lower $T_c^*$. We
delve deeper into the intricate interplay between Hund's rule coupling and
inter-orbital hybridization by tuning the two parameters, and reveal two
distinct SC phases dominated by Hund's rule and hybridization, respectively,
separated by a non-SC regime. Our results treat the two $e_g$ orbitals in a
bilayer model on an equal footing, and clarify the essential roles of Hund's
rule and hybridization in the formation of SC order in the pressurized
nickelate La$_3$Ni$_2$O$_7$.
