We present high-resolution spectroscopic measurements of the abundances of the α element titanium (Ti) and s-process elements yttrium (Y) and lanthanum (La) for 59 candidate M giant members of the Sagittarius (Sgr) dwarf spheroidal (dSph) + tidal tail system pre-selected on the basis of position and radial velocity (RV). As expected, the majority of these stars show peculiar abundance patterns compared to those of nominal Milky Way (MW) stars, but as a group, the stars form a coherent picture of chemical enrichment of the Sgr dSph from [Fe/H] = -1.4 to solar abundance. This sample of spectra provides the largest number of Ti, La, and Y abundances yet measured for a dSph, and spans metallicities not typically probed by studies of the other, generally more metal-poor MW satellites. On the other hand, the overall [Ti/Fe], [Y/Fe], [La/Fe], and [La/Y] patterns with [Fe/H] of the Sgr stream plus Sgr core do, for the most part, resemble those seen in the Large Magellanic Cloud (LMC) and other dSphs, only shifted by Δ[Fe/H] ~ +0.4 from the LMC and by ~+1 dex from the other dSphs; these relative shifts reflect the faster and/or more efficient chemical evolution of Sgr compared to the other satellites, and show that Sgr has had an enrichment history more like the LMC than the other dSphs. By tracking the evolution of the abundance patterns along the Sgr stream we can follow the time variation of the chemical make-up of dSph stars donated to the Galactic halo by Sgr. This evolution demonstrates that while the bulk of the stars currently in the Sgr dSph is quite unlike those of the Galactic halo, an increasing number of stars farther along the Sgr stream have abundances like MW halo stars, a trend that shows clearly how the Galactic halo could have been contributed by present-day satellite galaxies even if the present chemistry of those satellites is now different from typical halo field stars. Finally, we analyze the chemical abundances of a moving group of M giants among the Sgr leading arm stars at the North Galactic Cap, but having RVs unlike the infalling Sgr leading arm debris there. Through use of "chemical fingerprinting," we conclude that these mostly receding northern hemisphere M giants also are Sgr stars, likely trailing arm debris overlapping the Sgr leading arm in the north.