We study the effect of starting point and self-consistency within GW on the band-edge positions of semiconductors and insulators. Compared to calculations based on a semilocal starting point, the use of a hybrid-functional starting point shows a larger quasiparticle correction for both band-edge states. When the self-consistent treatment is employed, the band-gap opening is found to result mostly from a shift of the valence-band edge. Within the non-self-consistent methods, we analyse the performance of empirical and nonempirical schemes in which the starting point is optimally tuned. We further assess the accuracy of the band-edge positions through the calculation of ionization potentials of surfaces. The ionization potentials for most systems are reasonably well described by one-shot calculations. However, in the case of TiO2, we find that the use of self-consistency is critical to obtain a good agreement with experiment.