Blue supergiants (BSGs) are important objects to study the intermediate phases of massive star evolution, helping to constrain evolutionary models. However, the lack of a holistic study of a statistically significant and unbiased sample of these objects makes several long-standing questions about their nature remain unsolved. The present and other upcoming papers of the IACOB series are focused in studying - from a pure empirical point of view - a sample of 500 Galactic O9 - B9 stars with luminosity classes I and II (plus 250 late O- and early B-type stars with luminosity classes III, IV and V) and covering distances up to 4 kpc from the Sun. We compile an initial set of 11000 high-resolution spectra from 1600 Galactic late O- and B-type stars. We use a new novel spectroscopic strategy based on a simple fitting of the Hbeta line to select stars in a specific region of the spectroscopic HR diagram. We evaluate the completeness of our sample using the Alma Luminous Star catalog (ALS III) and Gaia-DR3 data. We show the benefits of the proposed strategy for identifying BSGs descending from stellar objects born as O-type stars, in the context of single star evolution. The resulting sample reaches a high level of completeness with respect to the ALS III catalog, gathering 80% of all-sky targets brighter than Bmag < 9 located within 2 kpc. However, we identify the need for new observations in specific regions of the Southern hemisphere. In conclusion, we have explored a very fast and robust method to select BSGs, providing a valuable tool for large spectroscopic surveys like WEAVE-SCIP or 4MIDABLE-LR, and highlighting the risk of using spectral classifications from the literature. Upcoming studies will make use of this large and homogeneous spectroscopic sample to study specific properties of these stars in detail. We initially provide first results about their rotational properties.