The NASA Transiting Exoplanet Survey Satellite (NASA-TESS) mission presents a treasure trove for understanding the stars it observes and the Milky Way, in which they reside. We present a first look at the prospects for Galactic and stellar astrophysics by performing initial asteroseismic analyses of bright (G < 11) red giant stars in the TESS southern continuous viewing zone (SCVZ). Using three independent pipelines, we detect νmax and Δν in 41 per cent of the 15 405 star parent sample (6388 stars), with consistency at a level of $\sim \! 2{{\ \rm per\ cent}}$ in νmax and $\sim \! 5{{\ \rm per\ cent}}$ in Δν. Based on this, we predict that seismology will be attainable for ∼3 × 105 giants across the whole sky and at least 104 giants with ≥1 yr of observations in the TESS-CVZs, subject to improvements in analysis and data reduction techniques. The best quality TESS-CVZ data, for 5574 stars where pipelines returned consistent results, provide high-quality power spectra across a number of stellar evolutionary states. This makes possible studies of, for example, the asymptotic giant branch bump. Furthermore, we demonstrate that mixed ℓ = 1 modes and rotational splitting are cleanly observed in the 1-yr data set. By combining TESS-CVZ data with TESS-HERMES, SkyMapper, APOGEE, and Gaia, we demonstrate its strong potential for Galactic archaeology studies, providing good age precision and accuracy that reproduces well the age of high [α/Fe] stars and relationships between mass and kinematics from previous studies based on e.g. Kepler. Better quality astrometry and simpler target selection than the Kepler sample makes this data ideal for studies of the local star formation history and evolution of the Galactic disc. These results provide a strong case for detailed spectroscopic follow-up in the CVZs to complement that which has been (or will be) collected by current surveys.