Excitation functions for total reaction cross sections, σR, were measured for the light, mainly proton-rich nuclei Li6, Be7, B10, C9,10,11, N12, O13,15, and Ne17 incident on a Si telescope at energies between 15 and 53 MeV/nucleon. The telescope served as target, energy degrader and detector. Proton-removal cross sections, σ2p for Ne17 and σp for most of the other projectiles, were also measured. The strong absorption model reproduces the A-dependence of σR, but not the detailed structure. Glauber multiple scattering theory and the Jeukenne, Lejeune, and Mahaux (JLM) folding model provided improved descriptions of the measured σR values. Rms radii, extracted from the measured σR using the optical limit of Glauber theory, are in good agreement with those obtained from high energy data. One-proton removal reactions are described using an extended Glauber model, incorporating second order noneikonal corrections, realistic single particle densities, and spectroscopic factors from shell model calculations.