This paper will present a first-order conservation law framework for large strain viscoelastic solid mechanics, in which the linear momentum conservation equation is combined with the geometric conservation laws and evolution equations for internal state variables. Two viscoelastic models will be considered. The first will address compressible viscoelasticity, extending our previously proposed strain energy function to incorporate time-dependent viscous effects. This leads to a standard linear evolution equation for internal state variables, which will be solved analytically. The second model will focus on incompressible viscoelasticity. We will develop a formulation that ensures internal entropy production whilst enforcing incompressibility via a Lagrange multiplier approach, resulting in a new evolution equation for internal state variables. This evolution equation will facilitate the proof of hyperbolicity. Hyperbolicity will be demonstrated for both viscoelastic models, guaranteeing the existence of real wave speeds across all deformation states. Explicit expressions for the corresponding pressure and shear wave speeds will be derived and presented. These accurate wave speed evaluations will enable the determination of optimal time step sizes for explicit time integration and will ensure that sufficient numerical dissipation is introduced into the numerical scheme. From a spatial discretisation standpoint, we will extend our upwind Smoothed Particle Hydrodynamics (SPH) to viscoelasticity, ensuring semi-discrete satisfaction of the second law of thermodynamics. The time rate of the Hamiltonian will be utilised to introduce entropy-stable numerical stabilisation and to assess internal entropy production within the system. This approach will ensure the consistent behaviour of the system and allow separate monitoring of material and numerical dissipations. The proposed framework will be validated through a series of numerical examples, with the SPH scheme benchmarked against an in-house vertex-based finite volume implementation and the mixed-based Updated Reference Lagrangian SPH algorithm.