Context. Recent studies have uncovered evidence that the statistical properties of asteroids' physical parameters are a fundamental source of information on the physics of their collisions and evolution. The analysis of the spin rates and spin vector distributions helps us to understand the role of various known and new effects. The alignment of spin vectors and the correlation of spin rates are for the first time observed for ten members of the Koronis family. These unexpected non-random orientations of the spin axes and correlations of the spin rates, now known as Slivan states are interpreted in terms of a YORP effect and spin-orbit resonances. Aims. To study non-gravitational-effects, there appears to be a need for new observational campaigns devoted to determining the physical parameters of the asteroid families. Methods. We analysed the photometric observations of the asteroids, which are the most efficient method of studying asteroid physical parameters. Results. We report the results of a ten-year long observational survey of the light variations of objects in the Flora region. We present 544 individual lightcurves of 55 objects obtained at various observing geometries. These lightcurves yield new or refined synodic periods for 32 asteroids and confirm period determinations for 23 objects in our sample. To improve the statistics of the Flora family objects, we add to our dataset 91 objects with reliably determined periods. The distribution of rotation rates for the Flora family is non-Maxwellian at a confidence level of 94% and different from those of the Koronis and the Hungaria families. It seems to be consistent with the long-term influence of the YORP effect, although it is also indicative of a younger age for the Flora family compared to both the Koronis and the Hungaria families. Conclusions. Our new data is a foundation for the spin vector and shape determinations that will be the objectives of the second paper of the series. We search for spin vector and spin periods correlations in order to determine whether Slivan states exist in the Flora family. ©2012 ESO.