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From subatomic particles to macromolecules, quantum mechanics has a deep impact in understanding the interactions and nature of such constituents of matter. The paradoxical observations of entanglement in quantum systems is used in quantum computations. Ion traps are among the leading platforms which provide clean and highly controllable quantum systems. This thesis presents the study of quantum coherence aiming to devise and test coherent control with trapped ions in a Penning trap. Our apparatus contains one or more calcium ions that are confined in space using DC electric and magnetic fields. Doppler cooling and resolved sideband cooling of trapped ions are carried out to perform high fidelity quantum control. The ions are prepared in their motional ground state via a multi-stage resolved sideband cooling technique. The cooling methods are first verified through simulations. The axial mean phonon numbers for a single ion are measured to be 0.03 ± 0.01 and 0.012 ± 0.009 at 187 kHz and 420 kHz axial frequencies, respectively. The heating rate is measured to be 2.5 ± 0.5 phonons/s, which is consistent with the previous measurements in our system. Similarly, the radial motional modes of a single ion are cooled to 〈n_+ 〉 < 0.4 and 〈n_- 〉 < 0.6 from mean phonon numbers corresponding to each mode of around 100. Preliminary studies are carried out of the axial motional modes of a 1D chain and 2D planar crystals, extending sideband cooling to small ICC's in a Penning trap. Coherent control is examined in our system and the optical coherence time and motional coherence time are estimated to be approximately 1 ms and 0.5 s, respectively. Furthermore, a single ion is prepared outside the Lamb-Dicke regime via a sideband heating technique and coherent studies are performed in such a regime via a laser beam.