The Dark Energy Spectroscopic Instrument (DESI) survey will measure large-scale structures using quasars as direct tracers of dark matter in the redshift range 0.9 < z < 2.1 and using Lyα forests in quasar spectra at z > 2.1. We present several methods to select candidate quasars for DESI, using input photometric imaging in three optical bands (g, r, z) from the DESI Legacy Imaging Surveys and two infrared bands (W1, W2) from the Wide-field Infrared Survey Explorer. These methods were extensively tested during the Survey Validation of DESI. In this paper, we report on the results obtained with the different methods and present the selection we optimized for the DESI main survey. The final quasar target selection is based on a random forest algorithm and selects quasars in the magnitude range of 16.5 < r < 23. Visual selection of ultra-deep observations indicates that the main selection consists of 71% quasars, 16% galaxies, 6% stars, and 7% inconclusive spectra. Using the spectra based on this selection, we build an automated quasar catalog that achieves a fraction of true QSOs higher than 99% for a nominal effective exposure time of ~1000 s. With a 310 deg-2 target density, the main selection allows DESI to select more than 200 deg-2 quasars (including 60 deg-2 quasars with z > 2.1), exceeding the project requirements by 20%. The redshift distribution of the selected quasars is in excellent agreement with quasar luminosity function predictions.