The band-edge structure and band gap are key parameters for a functional chalcogenide semiconductor to its applications in optoelectronics, nanoelectronics, and photonics devices. Here, we firstly demonstrate the complete study of experimental band-edge structure and excitonic transitions of monoclinic digallium trisulfide (Ga2S3) using photoluminescence (PL), thermoreflectance (TR), and optical absorption measurements at low and room temperatures. According to the experimental results of optical measurements, three band-edge transitions of EA = 3.052 eV, EB = 3.240 eV, and EC = 3.328 eV are respectively determined and they are proven to construct the main band-edge structure of Ga2S3. Distinctly optical-anisotropic behaviors by orientation- and polarization-dependent TR measurements are, respectively, relevant to distinguish the origins of the EA, EB, and EC transitions. The results indicated that the three band-edge transitions are coming from different origins. Low-temperature PL results show defect emissions, bound-exciton and free-exciton luminescences in the radiation spectra of Ga2S3. The below-band-edge transitions are respectively characterized. On the basis of experimental analyses, the optical property of near-band-edge structure and excitonic transitions in the monoclinic Ga2S3 crystal is revealed.