The main objectives of this series of papers are: (1) to demonstrate the existence of serious mutual disagreements between established total (and other integrated) magitude scales for Virgo galaxies; (2) to attempt to quantify both the systematic and random errors present within these magnitude scales; (3) to investigate the origins of any large error uncovered; and thereby (4) to encourage the general adoption of rigorous total-magnitude measurement procedures by the astronomical community. The ramifications of the findings presented in this series of papers will be discussed in detail at a later date. In this paper, the first in the series, a self-consistent dataset of trustworthy total-magnitude measurements is compiled for a sample of Virgo galaxies spanning a range of 10,000 in apparent brightness, based on only the most reliable measurements and photometry currently available. This reference dataset, which includes luminosity-profile shape information, will be used in subsequent papers as one of the bases for assessing existing magnitude scales for Virgo galaxies. As most published magnitudes are based on B-band observations, this series of papers will also focus primarily on B-band measurements.