The coupled Volterra lattice equation associated with $4 \times 4$ Lax pair is under investigation, which is an integrable discrete form of a coupled KdV equation applied widely in fluids, Bose–Einstein condensation and atmospheric dynamics. First, we explore the conditions for modulational instability (MI) of the constant seed background for this equation. Secondly, we present the discrete Darboux transformation (DT) and generalised DT based on the new $4 \times 4$ Lax pair. Through the resulting discrete DT, the bell-shaped and anti-$N$-shaped soliton solutions of the coupled Volterra lattice equation are derived. Moreover, we derive the $M$-shaped and $N$-shaped rational solitons and bell-shaped and $N$-shaped semirational soliton solutions of the coupled Volterra lattice equation via the discrete generalised DT. Finally, we numerically study the dynamical behaviours of such soliton solutions and find that the rational and semirational soliton solutions have better numerical stability than the usual soliton solution, although three types of solutions are robust against a small noise. The results may be helpful for understanding the two-layered fluid waves near ocean shores described by the coupled Korteweg–de Vries (KdV) equation.