Aims. One of the goals of the Gaia-ESO Survey (GES), which is conducted with FLAMES at the VLT, is the census and the characterization of the low-mass members of very young clusters and associations. We conduct a comparative study of the main properties of the sources belonging to γ Velorum (γ Vel) and Chamaeleon I (Cha I) young associations, focusing on their rotation, chromospheric radiative losses, and accretion. Methods. We used the fundamental parameters (effective temperature, surface gravity, lithium abundance, and radial velocity) delivered by the GES consortium in the first internal data release to select the members of γ Vel and Cha I among the UVES and GIRAFFE spectroscopic observations. A total of 140γ Vel members and 74 Cha I members were studied. The procedure adopted by the GES to derive stellar fundamental parameters also provided measures of the projected rotational velocity (vsini). We calculated stellar luminosities through spectral energy distributions, while stellar masses were derived by comparison with evolutionary tracks. The spectral subtraction of low-activity and slowly rotating templates, which are rotationally broadened to match the vsini of the targets, enabled us to measure the equivalent widths (EWs) and the fluxes in the Hα and Hβ lines. The Hα line was also used for identifying accreting objects, on the basis of its EW and the width at the 10% of the line peak (10%W), and for evaluating the mass accretion rate (M acc). Results. The distribution of vsini for the members of γ Vel displays a peak at about 10 km? s-1 with a tail toward faster rotators. There is also some indication of a different vsini distribution for the members of its two kinematical populations. Most of these stars have Hα fluxes corresponding to a saturated activity regime. We find a similar distribution, but with a narrower peak, for Cha? I. Only a handful of stars in γ Vel display signatures of accretion, while many more accretors were detected in the younger Cha I, where the highest Hα fluxes are mostly due to accretion, rather than to chromospheric activity. Accreting and active stars occupy two different regions in a Teff-flux diagram and we propose a criterion for distinguishing them. We derive M acc in the ranges 10-11-10-9 M⊙? yr-1 and 10-10-10-7 M⊙? yr-1 for γ Vel and Cha I accretors, respectively. We find less scatter in the M acc - M∗ relation derived through the Hα EWs, when compared to the Hα10%W diagnostics, in agreement with other authors.