  In this series of two papers we describe a project with STIS on HST to
measure the line-of-sight velocities of stars in the central few arcsec of the
dense globular cluster M15. This first paper focuses on the observations and
reduction of the data. We `scanned' the central region of M15 spectroscopically
by consecutively placing an 0.1 arcsec slit at 18 adjacent positions. Careful
and complicated data reduction and analysis were required. The spectral pixel
size exceeds the velocity dispersion of M15, which puts the project at the
limit of what is feasible with STIS. We applied corrections for the following
effects: drifts in the STIS wavelength scale during an HST orbit; the orbital
velocity component of HST along the line-of-sight to the cluster, and its
variations during the HST orbit; and the apparent wavelength shift that is
perceived for a star that is not centered in the slit. To make these
corrections we present a detailed study of the HST/STIS point-spread function
and line-spread function. We also present a catalog with positions and U, B and
V magnitudes of 31,983 stars in M15, created from existing HST/WFPC2 data. We
extract 19,200 one-dimensional spectra from the STIS data, each for a different
position in M15, and each with a velocity scale accurate to better than 2.5
km/s. We develop an algorithm that co-adds the spectra for individual apertures
and use it to extract spectra of individual stars with minimum blending and
maximum S/N. In Paper II we use these spectra to extract the line-of-sight
velocities of individual stars. The velocities constrain the central structure,
dynamics and mass distribution of the cluster, and provide evidence for the
existence of an intermediate-mass black hole in the center of the
cluster.[ABRIDGED]
