Context. Understanding stellar activity in M dwarfs is crucial for the physics of stellar atmospheres and for ongoing radial velocity exoplanet programmes. Despite the increasing interest in M dwarfs, our knowledge of the chromospheres of these stars is far from being complete. Aims: We test whether the relations between activity, rotation, and stellar parameters and flux-flux relationships previously investigated for main-sequence FGK stars and for pre-main-sequence M stars also hold for early-M dwarfs on the main-sequence. Although several attempts have been made so far, here we analyse a large sample of stars undergoing relatively low activity. Methods: We analyse in a homogeneous and coherent way a well-defined sample of 71 late-K/early-M dwarfs that are currently being observed in the framework of the HArps-N red Dwarf Exoplanet Survey (HADES). Rotational velocities are derived using the cross-correlation technique, while emission flux excesses in the Ca ii H & K and Balmer lines from Hα up to Hɛ are obtained by using the spectral subtraction technique. The relationships between the emission excesses and the stellar parameters (projected rotational velocity, effective temperature, kinematics, and age) are studied. Relations between pairs of fluxes of different chromospheric lines (flux-flux relationships) are also studied and compared with the literature results for other samples of stars. Results: We find that the strength of the chromospheric emission in the Ca ii H & K and Balmer lines is roughly constant for stars in the M0-M3 spectral range. Although our sample is likely to be biased towards inactive stars, our data suggest that a moderate but significant correlation between activity and rotation might be present, as well as a hint of kinematically selected young stars showing higher levels of emission in the calcium line and in most of the Balmer lines. We find our sample of M dwarfs to be complementary in terms of chromospheric and X-ray fluxes with those of the literature, extending the analysis of the flux-flux relationships to the very low flux domain. Conclusions: Our results agree with previous works suggesting that the activity-rotation-age relationship known to hold for solar-type stars also applies to early-M dwarfs. We also confirm previous findings that the field stars which deviate from the bulk of the empirical flux-flux relationships show evidence of youth. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.