  The TESS space telescope is collecting continuous, high-precision optical
photometry of stars throughout the sky, including thousands of RR Lyrae stars.
In this paper, we present results for an initial sample of 118 nearby RR Lyrae
stars observed in TESS Sectors 1 and 2. We use differential-image photometry to
generate light curves and analyse their mode content and modulation properties.
We combine accurate light curve parameters from TESS with parallax and color
information from the Gaia mission to create a comprehensive classification
scheme. We build a clean sample, preserving RR Lyrae stars with unusual light
curve shapes, while separating other types of pulsating stars. We find that a
large fraction of RR Lyrae stars exhibit various low-amplitude modes, but the
distribution of those modes is markedly different from those of the bulge
stars. This suggests that differences in physical parameters have an observable
effect on the excitation of extra modes, potentially offering a way to uncover
the origins of these signals. However, mode identification is hindered by
uncertainties when identifying the true pulsation frequencies of the extra
modes. We compare mode amplitude ratios in classical double-mode stars to stars
with extra modes at low amplitudes and find that they separate into two
distinct groups. Finally, we find a high percentage of modulated stars among
the fundamental-mode pulsators, but also find that at least 28% of them do not
exhibit modulation, confirming that a significant fraction of stars lack the
Blazhko effect.
