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. 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. Using a spectral subtraction technique, we measured pseudo-equivalent widths of the HeI D3, H-alpha, HeI 10833, and Pa-beta lines, the NaI D doublet, and the CaII 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. We find that the most slowly rotating stars of each spectral subtype have the strongest H-alpha absorption. H-alpha is correlated most strongly with HeI D3, whereas NaI D and the CaII infrared triplet are also correlated with H-alpha. HeI 10833 and Pa-beta 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 H-alpha and HeI D3 are smaller for stars with higher activity levels, while this anti-correlation is weaker for NaI D and the CaII infrared triplet, and is absent for HeI 10833 and Pa-beta. Periodic variation with the rotation period most commonly appears in the TiO bands, H-alpha, and in the CaII infrared triplet.