The different projects of this thesis contribute to improving our understanding of the early chemical evolution of the galaxies. The standard cosmological model states an hierarchical formation of the galaxies, the dwarf galaxies being the first stellar systems to form in the early Universe, merging to build larger galaxies over time. Observing the oldest stars in different environments can therefore answer many questions for the models of galaxy formation. Do the old stellar populations of the dwarf galaxies share the same chemical abundance patterns as the Milky Way halo ? These oldest and extremely metal-poor stars (EMP, [Fe/H]<-3) are challenging to find and to observe, especially outside our own Galaxy. Less than 10 EMP stars have very detailed chemical analysis within each of the closest dwarf galaxies, the Sculptor, Fornax, Carina and Sextans dwarf spheroidal galaxies (dSphs). During a part of this thesis, detailed spectroscopic analyses have being carried at high resolution for different sets of stars observed with the Very Large Telescope in 3 dwarf spheroidal galaxies : Sextans, Fornax and Carina. I derived the abundances of a large set of chemical elements : the alpha-element group (Mg, Si, Ca, Ti), iron-peak (Sc, Cr, Fe, Co, Ni, Mn, Zn) and neutron-capture (Y, Sr, Ba, Eu) elements, providing the very first overview for some of them. I showed that Sextans presents an alpha-rich plateau in its early history, compatible with the Milky Way (MW) halo composition, contrary to what was previously suggested. Fornax and Carina barely explored in their EMP regime have now their alpha-plateau identified. Another part of this thesis focused on the analysis of a sample of 132 candidate metal-poor stars, observed by the Pristine survey in the Milky Way. Only 1/2000 star is expected to be extremely metal-poor if observations are done blindly, and the statistics become much lower for the ultra metal-poor stars (UMP, [Fe/H]<-4). During the last decades, few hundreds of EMP stars, and ~40 UMP stars, have been found by dedicated surveys mainly in the Southern hemisphere. The Pristine survey was created few years ago to cover the Northern hemisphere, and has already proved the very high efficiency of its photometric calibration in finding these rare targets. A first large sample of the brightest metal-poor candidates has been analysed at high resolution. I delivered the first general view on the chemical evolution of the MW as witnessed by the Pristine survey.