  (Abridged) Understanding stellar activity in M dwarfs is crucial for the
physics of stellar atmospheres as well as 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. We aim to test whether
the relations between activity, rotation, and stellar parameters and flux-flux
relationships also hold for early-M dwarfs on the main-sequence. We analyse in
an 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 Halpha up to Hepsilon are obtained by using the spectral
subtraction technique. The relationships between the emission excesses and the
stellar parameters are studied. Relations between pairs of fluxes of different
chromospheric lines are also studied. 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. 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. 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. 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.
