We present 12 epochs of optical spectroscopy taken across the discovery outburst of the black hole (BH) candidate MAXI J1803-298 with the Gran Telescopio Canarias and Very Large Telescope. The source followed a standard outburst evolution with hard and soft states. The system displays a triangular shape in the hardness intensity diagram, consistent with that seen in high-inclination BH transients and the previously reported detection of X-ray dips. The two epochs observed during the initial hard state exhibited asymmetric emission-line profiles, including a P-Cygni profile simultaneously detected in Hα and He I 6678, which indicates the presence of an optical wind in the system. The remaining spectra, obtained during the transition to the soft state and the subsequent decay, are instead characterized by narrower, double-peaked emission lines embedded into broad absorption components. One epoch (intermediate state) also includes near-infrared (NIR) coverage, revealing complex line profiles in the Paschen and Bracket series, which suggests that the outflow is still present during the outburst decay through the soft state. The growing list of low-mass X-ray binaries with optical and NIR outflow signatures indicates that these are common features. Furthermore, the lowest luminosity spectrum exhibits an Hα FWHM of 1570 ± 100 km s-1. This, together with previous constraints on the binary parameters, allows us to favor a compact object mass of ~3-10 M ⊙, further supporting its BH nature.