Blue supergiants (BSGs) are key objects for understanding the evolution of massive stars. However, discrepancies between theoretical predictions and empirical observations have opened up important questions yet to be answered. Studying statistically significant and unbiased samples of BSGs can help to improve the situation. We aim to perform a homogeneous and comprehensive quantitative spectroscopic analysis of a large sample of Galactic BSGs from the IACOB spectroscopic database. We derive the projected rotational velocity (vsini) and macroturbulent broadening (vmac) using IACOB-BROAD. We used FASTWIND computations to derive effective temperatures (Teff), surface gravities, microturbulences (ξ), Si and He surface abundances, and the wind-strength parameter. We provide estimates of the above-mentioned quantities for the largest sample of Galactic BSGs spectroscopically analyzed to date, comprising 538 O9-B5 type stars. We find a drop in the relative number of BSGs at ~21 kK, coinciding with a scarcity of fast rotating stars below that temperature. We speculate that this feature might be roughly delineating the location of the empirical Terminal-Age-Main-Sequence in the 15-85Msol range. By investigating the main characteristics of the vsini distribution of O stars and BSGs as a function of Teff, we propose that an efficient mechanism transporting angular momentum from the stellar core to the surface might be operating along the main sequence. We find correlations between ξ, vmac and the spectroscopic luminosity. We also find that no more than 20% of the BSGs have atmospheres enriched in He, and suggest that the origin of this specific sub-sample of BSGs might be in binary evolution. We do not find clear empirical evidence of an increase in the wind-strength over the wind bi-stability region towards lower Teff.