ESPRESSO is a high-resolution-ultra-stable spectrograph for the Very Large Telescope (VLT), whose commissioning will start in 2017. One of its key science goals is to test the stability of nature’s fundamental couplings with unprecedented accuracy and control of possible systematics. A total of 27 nights of the ESPRESSO Consortium’s guaranteed time observations (GTO) will be spent on testing the stability of the fine-structure constant and other fundamental couplings. A set of 14 priority optimal targets have been selected for the GTO period. In this work, we discuss the criteria underlying this selection, describe the selected targets, and present some forecasts of the impact of these measurements on fundamental physics and cosmology, focusing on dark energy constraints and using future supernova type Ia surveys as a comparison point. This report is a summary of the results reported in Phys. Rev. D 2016, 94, 123512, to which we refer the reader for further details.