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We describe the target selection and characteristics of the DESI Peculiar Velocity Survey, the largest survey of peculiar velocities (PVs) using both the fundamental plane (FP) and the Tully–Fisher (TF) relationship planned to date. We detail how we identify suitable early-type galaxies (ETGs) for the FP and suitable late-type galaxies (LTGs) for the TF relation using the photometric data provided by the DESI Legacy Imaging Survey DR9. Subsequently, we provide targets for 373 533 ETGs and 118 637 LTGs within the Dark Energy Spectroscopic Instrument (DESI) 5-yr footprint. We validate these photometric selections using existing morphological classifications. Furthermore, we demonstrate using survey validation data that DESI is able to measure the spectroscopic properties to sufficient precision to obtain PVs for our targets. Based on realistic DESI fibre assignment simulations and spectroscopic success rates, we predict the final DESI PV Survey will obtain ∼133 000 FP-based and ∼53 000 TF-based PV measurements over an area of 14 000 deg2. We forecast the ability of using these data to measure the clustering of galaxy positions and PVs from the combined DESI PV and Bright Galaxy Surveys (BGS), which allows for cancellation of cosmic variance at low redshifts. With these forecasts, we anticipate a 4 per cent statistical measurement on the growth rate of structure at z < 0.15. This is over two times better than achievable with redshifts from the BGS alone. The combined DESI PV and BGS will enable the most precise tests to date of the time and scale dependence of large-scale structure growth at z < 0.15.