We report one of the most accurate measurements of the three-dimensional large-scale galaxy power spectrum achieved to date, using 56 159 redshifts of bright emission-line galaxies at effective redshift z≈ 0.6 from the WiggleZ Dark Energy Survey at the Anglo-Australian Telescope. We describe in detail how we construct the survey selection function allowing for the varying target completeness and redshift completeness. We measure the total power with an accuracy of approximately 5 per cent in wavenumber bands of Δk= 0.01 h Mpc−1 . A model power spectrum including non-linear corrections, combined with a linear galaxy bias factor and a simple model for redshift-space distortions, provides a good fit to our data for scales k < 0.4 h Mpc−1 . The large-scale shape of the power spectrum is consistent with the best-fitting matter and baryon densities determined by observations of the cosmic microwave background radiation. By splitting the power spectrum measurement as a function of tangential and radial wavenumbers, we delineate the characteristic imprint of peculiar velocities. We use these to determine the growth rate of structure as a function of redshift in the range 0.4 < z < 0.8 , including a data point at z= 0.78 with an accuracy of 20 per cent. Our growth rate measurements are a close match to the self-consistent prediction of the Λ cold dark matter model. The WiggleZ survey data will allow a wide range of investigations into the cosmological model, cosmic expansion and growth history, topology of cosmic structure and Gaussianity of the initial conditions. Our calculation of the survey selection function will be released at a future date via our website wigglez.swin.edu.au.