We perform a systematic investigation of variational forms (wave-function Ansätze), to determine the ground-state energies and properties of two-dimensional model fermionic systems on triangular lattices (with and without periodic boundary conditions), using the variational quantum eigensolver (VQE) algorithm. In particular, we focus on the nature of the entangler blocks which provide the most efficient convergence to the system ground state inasmuch as they use the minimal number of gate operations, which is key for the implementation of this algorithm in noisy intermediate-scale quantum computers. Using the concurrence measure, the amount of entanglement of the register qubits is monitored during the entire optimization process, illuminating its role in determining the efficiency of the convergence. Finally, we investigate the scaling of the VQE circuit depth as a function of the desired energy accuracy. We show that the number of gates required to reach a solution within an error ɛ follows the Solovay-Kitaev scaling, O[log10c(1/ɛ)], with an exponent c=1.31±0.13.