The Transiting Exoplanet Survey Satellite (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 analyze 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.