We present the first results with the ROentgen Survey with an Imaging Telescope Array (eROSITA) on board the Russian Spektrum-Roentgen-Gamma mission (SRG), and we combine the new X-ray data with observations with the Transiting Exoplanet Survey Satellite (TESS). We use the superblink proper motion catalog of nearby M dwarfs as input sample to search for eROSITA and TESS data. We extracted Gaia-DR2 data for the full M dwarf catalog that comprises ~9000 stars, and we calculated the stellar parameters from empirical relations with optical/IR colors. Then we cross-matched this catalog with the eROSITA Final Equatorial Depth Survey (eFEDS) and the first eROSITA all-sky survey (eRASS1). After a meticulous source identification in which we associate the closest Gaia source to the eROSITA X-ray detections, our sample of M dwarfs is defined by 687 stars with SpT = K5..M7 (673 from eRASS1 and 14 from eFEDS). While for eRASS1 we used the data from the source catalog provided by the eROSITA_DE consortium, for the much smaller eFEDS sample we performed the data extraction and we analyzed the X-ray spectra and light curves. This unprecedented data base for X-ray emitting M dwarfs allowed us to put a quantitative constraint on the mass dependence of the X-ray luminosity, and to determine the change in the activity level with respect to pre-main-sequence stars. TESS observations are available for 489 of 687 X-ray detected M dwarfs, and applying standard period search methods we could determine the rotation period for 180 X-ray detected M dwarfs, about one forth of the X-ray sample. With the joint eROSITA and TESS sample, and combining it with our compilation of historical X-ray and rotation data for M dwarfs, we examine the mass dependence of the "saturated" regime of the rotation-activity relation. A first comparison of eROSITA hardness ratios and spectra shows that 65% of the X-ray detected M dwarfs have coronal temperatures of ~0.5keV. We performed a statistical investigation of the long-term X-ray variability of M dwarfs comparing the eROSITA measurements to those obtained ~30 years earlier during the ROSAT all-sky survey (RASS). Evidence for X-ray flares are found in various parts of our analysis: directly from an inspection of the eFEDS light curves, in the relation between RASS and eRASS1 X-ray luminosities, and in a subset of stars that displays X-ray emission hotter than the bulk of the sample according to the hardness ratios. Finally, we point out the need to obtain X-ray spectroscopy for more M dwarfs to study the coronal temperature-luminosity relation that is not well constrained by our eFEDS results.