  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)
