Trapped ions are one of the best candidate systems to realize quantum computation. In our laboratory, we are trying to implement quantum computing and information processing: two hyperfine ground-states of magnesium-25 ions will serve as the two-level system to store quantum information. The ions are confined in a linear radio-frequency trap under ultra-high vacuum conditions and will be cooled down to their motional ground-states. By illuminating the ions with frequency-stabilized lasers we will be able to initialize, manipulate, and read out their internal electronic quantum states in a well-controlled way and with high fidelity. In addition, the ions can be made to interact with each other by coupling their internal electronic states to a collective vibrational mode of motion along the trap axis. In this thesis, the focus will be on the process of building a trapped-magnesium-ion quantum information processor.