  We present the first asteroseismic results for $\delta$ Scuti and $\gamma$
Doradus stars observed in Sectors 1 and 2 of the TESS mission. We utilise the
2-min cadence TESS data for a sample of 117 stars to classify their behaviour
regarding variability and place them in the Hertzsprung-Russell diagram using
Gaia DR2 data. Included within our sample are the eponymous members of two
pulsator classes, $\gamma$ Doradus and SX Phoenicis. Our sample of pulsating
intermediate-mass stars observed by TESS also allows us to confront theoretical
models of pulsation driving in the classical instability strip for the first
time and show that mixing processes in the outer envelope play an important
role. We derive an empirical estimate of 74% for the relative amplitude
suppression factor as a result of the redder TESS passband compared to the
Kepler mission using a pulsating eclipsing binary system. Furthermore, our
sample contains many high-frequency pulsators, allowing us to probe the
frequency variability of hot young $\delta$ Scuti stars, which were lacking in
the Kepler mission data set, and identify promising targets for future
asteroseismic modelling. The TESS data also allow us to refine the stellar
parameters of SX Phoenicis, which is believed to be a blue straggler.
