The large bulk bandgap (1.35 eV) and Bohr radius (~10 nm) of InP semiconductor nanocrystals provides bandgap tunability over a wide spectral range, providing superior color tuning compared to that of CdSe quantum dots. In this paper, the dependence of the bandgap, photoluminescence emission, and exciton radiative lifetime of core/shell quantum dot heterostructures has been investigated using colloidal InP core nanocrystals with multiple diameters (1.5, 2.5, and 3.7 nm). The shell thickness and composition dependence of the bandgap for type-I and type-II heterostructures was observed by coating the InP core with ZnS, ZnSe, CdS, or CdSe through one to ten iterations of a successive ion layer adsorption and reaction (SILAR)-based shell deposition. The empirical results are compared to bandgap energy predictions made with effective mass modeling. Photoluminescence emission colors have been successfully tuned throughout the visible and into the near infrared (NIR) wavelength ranges for type-I and type-II heterostructures, respectively. Based on sizing data from transmission electron microscopy (TEM), it is observed that at the same particle diameter, average radiative lifetimes can differ as much as 20-fold across different shell compositions due to the relative positions of valence and conduction bands. In this direct comparison of InP/ZnS, InP/ZnSe, InP/CdS, and InP/CdSe core/shell heterostructures, we clearly delineate the impact of core size, shell composition, and shell thickness on the resulting optical properties. Specifically, Zn-based shells yield type-I structures that are color tuned through core size, while the Cd-based shells yield type-II particles that emit in the NIR regardless of the starting core size if several layers of CdS(e) have been successfully deposited. Particles with thicker CdS(e) shells exhibit longer photoluminescence lifetimes, while little shell-thickness dependence is observed for the Zn-based shells. Taken together, these InP-based heterostructures demonstrate the extent to which we are able to precisely tailor the material properties of core/shell particles using core/shell dimensions and composition as variables.