Laser-driven trapped ions are one of the most developed quantum technologies to date. Their applicability ranges from experimentally testing of basic quantum principles to the physical realisation of quantum computers and simulators. Usually, these systems have been operated in regimes of low laser intensity. This regime allows the use of simple effective Hamiltonians based on the validity of the rotating-wave approximation. The advent of novel trap and cavity designs that offer the possibility of strongly driving the ions opens up new avenues for the implementation of faster quantum operations and the emergence of new physics through nonlinear effects. From the theoretical side, however, there is still much to be explored in strong-driving scenarios. The aim of this thesis, then, is to investigate and extend our understanding of trapped ion systems beyond the low-intensity regime. Our efforts and contributions can be split into two fronts. First, with the use of specific unitary transformations, we proposed a model that effectively describes the trapped ion dynamics from weak-driving scenarios up to strong-driving ones. Besides, it also allows the description of strong-driving dynamics even when the laser is not in perfect tune with the atomic transition. Second, we explored and numerically characterised the nonequilibrium dynamics of a trapped ion subject to a temperature gradient. In particular, we studied how properties such as heat currents, steady-state residual coherences, and current rectification, which are of interest to several quantum information and thermodynamical applications, respond to changes in the trapped ion parameters, with a particular focus in high-intensity scenarios.