Context. The distribution of exoplanets around low-mass stars is still not well understood. Such stars, however, present an excellent opportunity for reaching down to the rocky and habitable planet domains. The number of current detections used for statistical purposes remains relatively modest and different surveys, using both photometry and precise radial velocities, are searching for planets around M dwarfs. Aims: Our HARPS-N red dwarf exoplanet survey is aimed at the detection of new planets around a sample of 78 selected stars, together with the subsequent characterization of their activity properties. Here we investigate the survey performance and strategy. Methods: From 2700 observed spectra, we compare the radial velocity determinations of the HARPS-N DRS pipeline and the HARPS-TERRA code, calculate the mean activity jitter level, evaluate the planet detection expectations, and address the general question of how to define the strategy of spectroscopic surveys in order to be most efficient in the detection of planets. Results: We find that the HARPS-TERRA radial velocities show less scatter and we calculate a mean activity jitter of 2.3 m s-1 for our sample. For a general radial velocity survey with limited observing time, the number of observations per star is key for the detection efficiency. In the case of an early M-type target sample, we conclude that approximately 50 observations per star with exposure times of 900 s and precisions of approximately 1 ms-1 maximizes the number of planet detections. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias (IAC).