Context. Gaia DR3, released in June 2022, included low-resolution XP spectra that have been used for the classification of various types of emission-line objects through machine-learning techniques. The Gaia Extended Stellar Parametrizer for Emission-Line Stars (ESP-ELS) algorithm identified 565 sources as potential Wolf-Rayet (WR) stars. Over half of them were already known as WR stars in the Milky Way and Magellanic Clouds. Aims. We utilized Gaia DR3 data to identify new Galactic WR stars. Methods. We extracted all sources from the Gaia catalog classified as WC- or WN-type stars by the ESP-ELS algorithm. By applying judicious 2MASS color selection criteria, leveraging Gaia Hα measurements, and filtering out objects already cataloged in various databases, we selected 37 bright candidates (G ≤ 16 mag) and 22 faint candidates (G > 16 mag). Spectroscopic follow-up observations of these candidates were conducted using the 2SPOT facilities in Chile and France, as well as the 1 m C2PU's Epsilon telescope at the Calern Observatory. Results. This paper focuses on the brighter sample. Among the 37 targets, we confirmed 17 and 16 new Galactic WC- and WN-type WR stars, respectively. Three of them were recently reported as new WR stars in an independent study. Conclusions. The Gaia mission provides a valuable resource for identifying WR stars missed in earlier surveys. While this work concentrated on a relatively small starting sample provided by the ESP-ELS algorithm, our findings highlight the potential for refining selection criteria to identify additional candidates not included in the outputs of the algorithm. Furthermore, the observation program underscores the utility of small telescopes in acquiring initial spectral data for sources with magnitudes up to G ~ 16 mag.