We have commenced a multi-year program, the Caltech-NRAO Stripe 82 Survey (CNSS), to search for radio transients with the Jansky VLA in the SDSS Stripe 82 region. The CNSS will deliver five epochs over the entire $\sim$270 deg$^2$ of Stripe 82, an eventual deep combined map with a rms noise of $\sim$40 $\mu$Jy 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 deg$^2$ region of Stripe 82, involving four epochs spanning logarithmic timescales between one week and 1.5 years, with the combined map having a median rms noise of 35 $\mu$Jy. 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 deg$^2$ survey. Classification of variable and transient sources relied heavily on the wealth of multi-wavelength data in the Stripe 82 region, supplemented by repeated mapping of the region by the Palomar Transient Factory. $3.9^{+0.5}_{-0.9}$% of the detected point sources were found to vary by greater than 30%, consistent with similar studies at 1.4 GHz and 5 GHz. Multi-wavelength photometric data and light curves suggest that the variability is mostly due to shock-induced flaring in the jets of AGN. 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 radio survey data revealed additional highly variable and transient sources on timescales between 5-20 years, largely associated with renewed AGN activity. The rates of such AGN possibly imply episodes of enhanced accretion and jet activity occurring once every $\sim$40,000 years in these galaxies. We compile the revised radio transient rates and make recommendations for future transient surveys and joint radio-optical experiments. (Abridged)
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