What are the advantages of radio spectroscopy? What is the main purpose for radio spectroscopy? What information of the atomic structure of atoms can we gain from using radio frequency radiation rather than X-rays? Since the wavelength of radio waves is on the meter scale, I am having trouble intuitively coming up with application specific to radio spectroscopy. I know that it is used to measure the Zeeman effect on atoms in the presence of a magnetic field but I don't fully understand why radio freq. is superior to other frequencies.
 A: There is nothing superior about using any particular sector of the electromagnetic spectrum, per se.
Different types of atomic transitions emit different photon energies, from X-rays to RF. RF spectroscopy will give you information about only those transitions that involve energy levels separated by RF-like energy levels.
For example, typical Zeeman shifts for ground states of alkali atoms for weak magnetic fields are around MHz, which fall within the Radio spectrum. So usually you talk about RF spectroscopy (to, for example, calibrate the magnetic field), or RF state transfer, etc.
On the other hand, the energy level separation between the ground state hyperfine structure of e.g. Rb or Cs is around $5-10$ GHz, which falls within the Microwave (MW) spectrum. So you'd talk about MW state transfers, MW spectroscopy, etc.
X-rays are very energetic and are usually associated with either nuclear transitions, or electron transitions to low energy levels in heavy atoms.
A: Radio spectroscopy is an astronomy tool which lets the spectroscopist detect the presence of certain molecules or atoms either at the origin of a distant radio beam or in the interstellar space between the beam source and the spectroscopic detector.  Since different types of radio sources have different characteristic spectra, studying the spectrum via radio spectroscopy lets us identify which type of source we are looking at.
It is also possible to combine the radio signals from a network of radio receivers on earth looking at the same distant source to create a virtual radio telescope, with an aperture thousands of kilometers across- far larger than that of any single optical telescope- so the spatial resolution of such a radio interferometer can be extremely high. This technique lets us assemble highly detailed "radio images" of very distant radio sources which would be impossible to obtain using visible light.
A: In case if x-rays we are dealing with diffraction spectroscopy, where the wave length is about, e.g., crystal spacing that we are imaging. In the case of optical or microwave radiation we have emission/absorption spectroscopy, where the frequency matches the absorption frequency of atoms, nuclei, etc. In particular, the energies of Zeeman splitting of electrons and nuclei correspond to microwave and radio range and studied using techniques, known as EPR (electron paramagnetic resonance) and NMR (nuclear magnetic resonance). The latter has broad application under the name of MRI (magnetic resonance imaging).
