At 66 Mpc, AT2019qiz is the closest optical tidal disruption event (TDE) to date, with a luminosity intermediate between the bulk of the population and the faint-and-fast event iPTF16fnl. Its proximity allowed a very early detection and triggering of multiwavelength and spectroscopic follow-up well before maximum light. The velocity dispersion of the host galaxy and fits to the TDE light curve indicate a black hole mass ≈106 M⊙, disrupting a star of ≈1 M⊙. By analysing our comprehensive UV, optical, and X-ray data, we show that the early optical emission is dominated by an outflow, with a luminosity evolution L ∝ t2, consistent with a photosphere expanding at constant velocity (≳2000 km s−1), and a line-forming region producing initially blueshifted H and He ii profiles with v = 3000–10 000 km s−1. The fastest optical ejecta approach the velocity inferred from radio detections (modelled in a forthcoming companion paper from K. D. Alexander et al.), thus the same outflow may be responsible for both the fast optical rise and the radio emission – the first time this connection has been observed in a TDE. The light-curve rise begins 29 ± 2 d before maximum light, peaking when the photosphere reaches the radius where optical photons can escape. The photosphere then undergoes a sudden transition, first cooling at constant radius then contracting at constant temperature. At the same time, the blueshifts disappear from the spectrum and Bowen fluorescence lines (N iii) become prominent, implying a source of far-UV photons, while the X-ray light curve peaks at ≈1041 erg s−1. Assuming that these X-rays are from prompt accretion, the size and mass of the outflow are consistent with the reprocessing layer needed to explain the large optical to X-ray ratio in this and other optical TDEs, possibly favouring accretion-powered over collision-powered outflow models.