Transition waves are central to phase-boundary propagation in solid phase transformations and sequential state switching in multistable metamaterials. However, existing studies are often limited to specific structural prototypes or simplified constitutive models, failing to reveal how wave kinetics and stability can be independently modulated by bistable element parameters. Here, we investigate transition waves in viscoelastic bistable lattices using a trilinear constitutive model with independently tunable parameters. Exact traveling wave solutions are obtained through the nonlinear map approach, while the quasicontinuum approximation provides analytical insight and efficient initial guesses for numerical continuation. Results indicate that the energy release rate, boundary strain, transition zone width, and wave profile are strongly regulated by damping and the stiffness characteristics of the bistable spring. Floquet analysis supports the proposed macroscopic criterion that instability arises when the energy release rate decreases with increasing wave velocity, and further reveals that the stable velocity region is mainly governed by the stiffnesses of both the untransformed and transformed phases. Time-domain simulations show that low-velocity waves are typically unstable, manifesting in two distinct post-instability behaviors: they either split into two separated linear waves and a pinned transition interface, or self-adjust toward a more stable propagation state. Finally, a Riemann-type initial value problem is considered to determine the initial strain conditions required to initiate transition waves at prescribed velocities. These results establish a direct link between bistable constitutive parameters, kinetic relations, stability boundaries, post-instability evolution, and initiation conditions, providing a theoretical basis for designing transition-wave-mediated multistable metamaterials.