Dwarf galaxies like Sagittarius (Sgr) provide a unique window into the early stages of galactic chemical evolution, particularly through their metal-poor stars. By studying the chemical abundances of stars in the Sgr core and tidal streams, we can gain insights into the assembly history of this galaxy and its early heavy element nucleosynthesis processes. We efficiently selected extremely metal-poor candidates in the core and streams for high-resolution spectroscopic analysis using metallicity-sensitive photometry from SkyMapper DR2 and Gaia DR3 XP spectra, and proper motions. We present a sample of 37 Sgr stars with detailed chemical abundances, of which we identify 10 extremely metal-poor ([Fe/H]<=-3.0) stars, 25 very metal-poor ([Fe/H]<=-2.0) stars, and two metal-poor ([Fe/H]<=-1.0) stars. This sample increases the number of extremely metal-poor Sgr stars analyzed with high-resolution spectroscopy by a factor of 5. Of these stars, 15 are identified as members of the Sgr tidal stream, while the remaining 22 are associated with the core. We derive abundances for up to 20 elements and identify no statistically significant differences between the element abundance patterns across the core and stream samples. Intriguingly, we identify stars that may have formed in ultrafaint dwarf galaxies that accreted onto Sgr, in addition to patterns of C and r-process elements distinct from the Milky Way halo. Over half of the sample shows a neutron-capture element abundance pattern consistent with the scaled solar pure r-process pattern, indicating early r-process enrichment in the Sgr progenitor.