  We investigate the frustrated $J_1$-$J_2$ Ising model with nearest-neighbor
interaction $J_1$ and next-nearest-neighbor interaction $J_2$ in two kinds of
generalized triangular lattices (GTLs) employing the Wang--Landau Monte Carlo
method and finite-size scaling analysis. In the first GTL (GTL1), featuring
anisotropic properties, we identify three kinds of super-antiferromagnetic
ground states with stripe structures. Meanwhile, in the second GTL (GTL2),
which is non-regular in next-nearest-neighbor interaction, the ferrimagnetic
3$\times$3 and two kinds of partial spin liquid ground states are observed. We
confirm that residual entropy is proportional to the number of spins in the
partial spin liquid ground states. Additionally, we construct
finite-temperature phase diagrams for ferromagnetic nearest-neighbor and
antiferromagnetic next-nearest-neighbor interactions. In GTL1, the transition
into the ferromagnetic phase is continuous, contrasting with the first-order
transition into the stripe phase. In GTL2, the critical temperature into the
ferromagnetic ground state decreases as antiferromagnetic next-nearest-neighbor
interaction intensifies until it meets the 3$\times$3 phase boundary. For
intermediate values of the next-nearest-neighbor interaction, two successive
transitions emerge: one from the paramagnetic phase to the ferromagnetic phase,
followed by the other transition from the ferromagnetic phase to the 3$\times$3
phase.
