The field of cold molecular ions is a fast growing one, with applications in high resolution spectroscopy and metrology, the search for time variations of fundamental constants, cold chemistry and collisions, and quantum information processing, to name a few. The study of single molecular ions is attractive as it enables one to push the limits of spectroscopic accuracy. Non-destructive spectroscopic detection of molecular ions can be achieved by co-trapping with an easier to detect atomic ion. The ion chain has coupled motion, and transitions which change both the internal and motional states of the molecule can be detected on the atomic ion as a change its motion. Scattered photons of the mid-IR rovibrational transitions in molecular ions are less energetic than those of the optical transitions in atomic ions, and are thus less likely to change the motional state of the ion crystal. This thesis proposes to adiabatically relax the trapping potential, called adiabatic cooling, when performing rovibrational excitations of the molecular ion to reduce the energy spacing of the harmonic motional levels, thus increasing the likelihood of a motional transition. The work presented in this thesis covers the implementation of adiabatic cooling for the application of rovibrational spectroscopy on single molecular ions. This entailed constructing and testing a new DC supply capable of employing adiabatic ramps of the ion's axial frequency on the 100's of us timescale. The DC supply went through several iterations to reduce the voltage noise output, which translates into unwanted motional excitation (heating) of the ions in the trap. Non-destructive detection relies on motional excitation coming only from optical excitation, thus it was necessary to characterize how the DC supply noise contributed to unwanted heating. Characterization of four different supply iterations was carried out for axial trap frequencies of a single Ca+ ranging from 68 kHz to 509 kHz, and reduction of the heating rates was observed for each successive improvement of the DC supply. The supply's output noise spectrum has been directly connected to the measured heating rates, almost perfectly fitting existing heating rate theory. Further, the same model successfully predicted the heating rates of the in-phase mode of a two-ion crystal, indicating that we can use it to predict the heating rates in experiments on molecule-atom chains. Adiabatic cooling experiments with total ramp times of 290 us, 20-times faster than previous experiments, have been subsequently carried out, and show that heating due to technical noise is the main limitation in how low we can cool the ions. Results with the out-of-phase mode of a two-ion crystal demonstrate cooling with an axial frequency reduction of a factor of 12, with a mode temperature reduction of at least a factor 7. This results in a mode temperature of about 800 nK at the bottom of the ramp. Lastly, by reducing DC supply noise, I show reduction of the axial frequency below 50 kHz with nearly no motional gain is possible with some optimization. Rovibrational transitions in 24MgH+ are only known to the 1.5 GHz level compared to their Hz-linewidths. Simulations for broadband spectroscopy aimed reducing this uncertainty are presented for a rovibrational transition in 24MgH+. This technique allows for illumination times much less than the lifetime of the state. Adiabatic cooling is shown to decrease illumination time further by an order of magnitude for attainable trap parameters. Finally, further applications to adiabatic cooling for cold chemistry, collisions, and vibration-sensing are presented.