A moderate-resolution spectroscopic survey of Fleming's sample of 54 X-ray selected M dwarfs with photometric distances less than 25 pc is presented. All the objects consist of one or two dMe stars, some being doubles or spectroscopic binaries. Radial and rotation velocities have been measured by fits to the Halpha profiles. Radial velocities have been measured by cross correlation. Artificial broadening of an observed spectrum has produced a relationship between Halpha FWHM and rotation speed, which we use to infer rotation speeds for the entire sample by measurement of the Halpha emission line. We find three ultrafast rotators ( UFRs, v sin i greater than or equal to 100 km s(-1)) and eight stars with 30 km s(-1) v sin i < 100 km s(-1). We find that the UFRs have quite variable emission and should be observed for photometric variability. Cross-correlation velocities measured for UFRs are shown to depend on rotation speed and the filtering used. The radial velocity dispersion of the sample is 17 km s(-1). A new double emission line spectroscopic binary with a period of 3.55 days has been discovered, RX J1547.4+4507, and another known one is in the sample, the Hyades member RX J0442.5+2027. Three other objects are suspected spectroscopic binaries, and at least six are visual doubles. The only star in the sample observed to have significant lithium happens to be a known TW Hya association member, TWA 8A. These results all show that there are a number of young (<10(8) yr) and very young (<10(7) yr) low-mass stars in the immediate solar neighborhood. The Hα activity strength does not depend on rotation speed. Our fast rotators are less luminous than similarly fast rotators in the Pleiades. They are either younger than the Pleiades or gained angular momentum in a different way.