Context. The Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Echelle Spectrographs (CARMENES) survey is searching for Earth-like planets orbiting M dwarfs using the radial velocity method. Studying the stellar activity of the target stars is important to avoid false planet detections and to improve our understanding of the atmospheres of late-type stars. Aims. In this work we present measurements of activity indicators at visible and near-infrared wavelengths for 331 M dwarfs observed with CARMENES. Our aim is to identify the activity indicators that are most sensitive and easiest to measure, and the correlations among these indicators. We also wish to characterise their variability. Methods. Using a spectral subtraction technique, we measured pseudo-equivalent widths of the He I D₃, Hα, He I λ10833 Å, and Pa β lines, the Na I D doublet, and the Ca II infrared triplet, which have a chromospheric component in active M dwarfs. In addition, we measured an index of the strength of two TiO and two VO bands, which are formed in the photosphere. We also searched for periodicities in these activity indicators for all sample stars using generalised Lomb-Scargle periodograms. Results. We find that the most slowly rotating stars of each spectral subtype have the strongest Hα absorption. Hα is correlated most strongly with He I D₃, whereas Na I D and the Ca II infrared triplet are also correlated with Hα. He I λ10833 Å and Paβ show no clear correlations with the other indicators. The TiO bands show an activity effect that does not appear in the VO bands. We find that the relative variations of Hα and He I D₃ are smaller for stars with higher activity levels, while this anti-correlation is weaker for Na I D and the Ca II infrared triplet, and is absent for He I λ10833 Å and Paβ. Periodic variation with the rotation period most commonly appears in the TiO bands, Hα, and in the Ca II infrared triplet.