The modification of the nature and size of bandgaps for III-V semiconductors is of strong interest for optoelectronic applications. Strain can be used to systematically tune the bandgap over a wide range of values and induce indirect-to-direct transition (IDT), direct-to-indirect transition (DIT), and other changes in bandgap nature. Here, we establish a predictive first-principles approach, based on density functional theory, to analyze the effect of uniaxial, biaxial, and isotropic strain on the bandgap. We show that systematic variation is possible. For GaAs, DITs are observed at 1.56% isotropic compressive strain and 3.52% biaxial tensile strain, while for GaP an IDT is found at 2.63% isotropic tensile strain. We additionally propose a strategy for the realization of direct-to-indirect transition by combining biaxial strain with uniaxial strain. Further transition points are identified for strained GaSb, InP, InAs, and InSb and compared to the elemental semiconductor silicon. Our analyses thus provide a systematic and predictive approach to strain-induced bandgap tuning in binary III-V semiconductors.