  We present the first analysis of Cepheid stars observed by the TESS space
mission in Sectors 1 to 5. Our sample consists of 25 pulsators: ten fundamental
mode, three overtone and two double-mode classical Cepheids, plus three Type II
and seven anomalous Cepheids. The targets were chosen from fields with
different stellar densities, both from the Galactic field and from the
Magellanic System. Three targets have 2-minute cadence light curves available
by the TESS Science Processing Operations Center: for the rest, we prepared
custom light curves from the full-frame images with our own differential
photometric FITSH pipeline. Our main goal was to explore the potential and the
limitations of TESS concerning the various subtypes of Cepheids. We detected
many low amplitude features: weak modulation, period jitter, and timing
variations due to light-time effect. We also report signs of non-radial modes
and the first discovery of such a mode in an anomalous Cepheid, the overtone
star XZ Cet, which we then confirmed with ground-based multicolor photometric
measurements. We prepared a custom photometric solution to minimize saturation
effects in the bright fundamental-mode classical Cepheid, $\beta$ Dor with the
lightkurve software, and we revealed strong evidence of cycle-to-cycle
variations in the star. In several cases, however, fluctuations in the
pulsation could not be distinguished from instrumental effects, such as
contamination from nearby sources which also varies between sectors. Finally,
we discuss how precise light curve shapes will be crucial not only for
classification purposes but also to determine physical properties of these
stars.
