We propose a slanting collision scheme for Compton scattering of a laser light against a relativistic electron beam. This scheme is suitable to generate an energy-tunable X/γ-ray source. In this paper, we present theoretical study and simulation of the spectral, spatial and temporal characteristics of such a source. We also describe two terms laser-Compton scattering (LCS) experiments at the 100 MeV Linac of Shanghai Institute of Applied Physics, where quasi-monochromatic LCS X-ray energy spectra with peak energies of ～ 30 keV are observed successfully. These preliminary investigations are carried out to understand the feasibility of developing an energy-tunable quasi-monochromatic X/γ-ray source, the future Shanghai Laser Electron Gamma Source.