Precision spectroscopy of lithium ions offers a unique research platform for exploring bound state quantum electrodynamics and investigating the structure of atomic nuclei. This paper overviews our recent efforts dedicated to the precision theoretical calculations and experimental measurements of the hyperfine splittings of ^6,7Li ⁺ions in the $\,^3{\rm{S}}_1$ and $\,^3{\rm{P}}_J$ states. In our theoretical research, we utilize bound state quantum electrodynamics to calculate the hyperfine splitting of the $\,^3{\rm{S}}_1$ and $\,^3{\rm{P}}_J$ states with remarkable precision, achieving an accuracy on the order of $m\alpha^6$. Using Hylleraas basis sets, we first solve the non-relativistic Hamiltonian of the three-body system to derive high-precision energy and wave functions. Subsequently, we consider various orders of relativity and quantum electrodynamics corrections by using the perturbation method, with accuracy of the calculated hyperfine splitting reaching tens of kHz. In our experimental efforts, we developed a low-energy metastable lithium-ion source that provides a stable and continuous ion beam in the $\,^3{\rm{S}}_1$ state. Using this ion beam, we utilize the saturated fluorescence spectroscopy to enhance the precision of hyperfine structure splittings of ⁷Li ⁺in the $\,^3{\rm{S}}_1$ and $\,^3{\rm{P}}_J$ states to about 100 kHz. Furthermore, by utilizing the optical Ramsey method, we obtain the most precise values of the hyperfine splittings of ⁶Li ⁺, with the smallest uncertainty of about 10 kHz. By combining theoretical calculations and experimental measurements, our team have derived the Zemach radii of the ^6,7Li nuclei, revealing a significant discrepancy between the Zemach radius of ⁶Li and the values predicted by the nuclear model. These findings elucidate the distinctive properties of the ⁶Li nucleus, promote further investigations of atomic nuclei, and advance the precise spectroscopy of few-electron atoms and molecules.