  We present in this paper the results of a comprehensive GEANT simulation of
the Gamma-Ray Imaging Detector (GRID) being developed for the AGILE space
astrophysics mission. The GRID is designed to be sensitive in the 30 MeV - 50
GeV energy range, with excellent imaging and timing capabilities, and with a
very large field-of-view (~3 sr). In this paper (Paper I) we present the GRID
baseline geometry, a model for the charged particle and albedo-photon
backgrounds for an equatorial orbit of 550-600 km altitude, and the main
results of the first level (Level-1 Trigger) on-board data processing. Our
simulations show that the GRID Level-1 data processing is expected to be
capable of decreasing by a factor of ~ 20 the charged particle background (from
\~2 kHz to below 100 Hz), and by a factor of ~ 30 the albedo-photon background.
The gamma-ray photon detection efficiency by the imaging GRID is simulated to
be particularly efficient, varying between 39% and 17% depending on photon
energies and incident directions.
