We have commenced a multiyear program, the Caltech-NRAO Stripe 82 Survey (CNSS), to search for radio transients with the Jansky VLA in the Sloan Digital Sky Survey Stripe 82 region. The CNSS will deliver five epochs over the entire ~270 deg2 of Stripe 82, an eventual deep combined map with an rms noise of ~40 μJy and catalogs at a frequency of 3 GHz, and having a spatial resolution of 3''. This first paper presents the results from an initial pilot survey of a 50 deg2 region of Stripe 82, involving four epochs spanning logarithmic timescales between 1 week and 1.5 yr, with the combined map having a median rms noise of 35 μJy. This pilot survey enabled the development of the hardware and software for rapid data processing, as well as transient detection and follow-up, necessary for the full 270 deg2 survey. Data editing, calibration, imaging, source extraction, cataloging, and transient identification were completed in a semi-automated fashion within 6 hr of completion of each epoch of observations, using dedicated computational hardware at the NRAO in Socorro and custom-developed data reduction and transient detection pipelines. Classification of variable and transient sources relied heavily on the wealth of multiwavelength legacy survey data in the Stripe 82 region, supplemented by repeated mapping of the region by the Palomar Transient Factory. A total of ${3.9}_{-0.9}^{+0.5}$% of the few thousand detected point sources were found to vary by greater than 30%, consistent with similar studies at 1.4 and 5 GHz. Multiwavelength photometric data and light curves suggest that the variability is mostly due to shock-induced flaring in the jets of active galactic nuclei (AGNs). Although this was only a pilot survey, we detected two bona fide transients, associated with an RS CVn binary and a dKe star. Comparison with existing legacy survey data (FIRST, VLA-Stripe 82) revealed additional highly variable and transient sources on timescales between 5 and 20 yr, largely associated with renewed AGN activity. The rates of such AGNs possibly imply episodes of enhanced accretion and jet activity occurring once every ~40,000 yr in these galaxies. We compile the revised radio transient rates and make recommendations for future transient surveys and joint radio-optical experiments.