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Quantum simulations provide a useful tool to study a broad range of problems in physics, chemistry, and biology. Trapped ions are a versatile quantum simulator and a main contender for a universal circuit model quantum computer. They provide a highly controllable quantum environment, which grants access to measure phenomena in regimes that are not otherwise accessible in nature. Here, our aim is to show particular examples of quantum physics simulations with a trapped ion quantum computer/simulator. On one hand, we will focus on the simulation of para-particle oscillators; that is, a parity-deformed harmonic oscillator characterized by an order parameter. These oscillators generalize the standard Fermi-Dirac and Bose-Einstein statistics associated with fermions and bosons to para-particles. We realize a method for simulating and characterizing these alternative particles using a trapped-ion experiment. The combination of the Jaynes-Cummings and anti-Jaynes Cummings dynamics present in a trapped ion coupled to multiple modes of motion simultaneously allows us to recover effective Hamiltonians which create a system analogous to para-Fermi or para-Bose oscillators. Trapped ions are a versatile quantum simulator and a main contender for a universal circuit model quantum computer. We use both of these flavors in this project, simulating para-Bosons digitally and para-Fermions directly by tailoring the native ion-mode couplings. We discuss the mapping steps and the latest experimental results. On the other hand, quantum walks provide a powerful framework for quantum simulation of physical systems. As a result, the community has been keen to see first implementations demonstrated, including single-particle relativistic quantum mechanics governed by the Dirac equation. Then, continuing with quantum circuit applications, our work presents a first programmable circuit implementation of quantum walks in one-dimensional space on a trapped-ion processor and its time-evolution up to five steps. The connection between quantum walks and Dirac cellular automata, a modeling framework for complex systems made up of simpler units, allows us to recover the characteristic spreading probability of an initially localized relativistic Dirac particle for different values of its mass. Due to a resource-efficient mapping of qubit states to walker positions, we achieve a high simulation performance despite going to deep quantum circuits (of up to 32 entangling gates).