  This is the first paper in a series presenting and analyzing data for a
K-selected sample of galaxies collected in order to identify and study galaxies
at moderate to high redshift in rest-wavelength optical light. The sample
contains 842 objects over 6 separate fields covering 75.6 arcmin^2 down to
K=20-20.5. We combine the K-band with UBVRIzJH multi-band imaging, reaching
depths of R~26. Two of the fields studied also have deep HST WFPC2 imaging,
totaling more than 60 hours in the F300W, F450W, F606W, and F814W filters.
Using artificial galaxy modeling and extraction we measure 85% completeness
limits down to K=19.5-20, depending on the field examined. The derived K-band
number counts are in good agreement with previous studies. We find a density
for Extremely Red Objects(EROs; R-K>5) of 1.55+/-0.16 arcmin^{-2} for K<19.7,
dominated by the 1714+5015 field (centered on 53w002), with an ERO number
density more than 3 times that of the other sample fields. If we exclude the
counts for 1714+5015, our density is 0.95+/- 0.14 arcmin. Both ERO densities
are consistent with previous measurements due to the significant known cosmic
variance of these red sources.Keck spectroscopic redshifts were obtained for 18
of the EROs, all but one of which are emission galaxies. None of the EROs in
the 1714+5015 field for which we obtained spectroscopic redshifts are
associated with the known z=2.39 over-density, although there are three
different galaxy redshift pairs (z=0.90, z=1.03, z=1.22).
