The Hubble Space Telescope has imaged the central part of M87 over a 10 week span, leading to the discovery of 32 classical novae and nine fainter, likely very slow and/or symbiotic novae. In this first in a series of papers we present the M87 nova finder charts, and the light and color curves of the novae. We demonstrate that the rise and decline times, and the colors of M87 novae are uncorrelated with each other and with position in the galaxy. The spatial distribution of the M87 novae follows the light of the galaxy, suggesting that novae accreted by M87 during cannibalistic episodes are well-mixed. Conservatively using only the 32 brightest classical novae we derive a nova rate for M87: $363_{-45}^{+33}$ novae/yr. We also derive the luminosity-specific classical nova rate for this galaxy, which is $7.88_{-2.6}^{+2.3} /yr/ 10^{10}L_\odot,_{K}$. Both rates are 3-4 times higher higher than those reported for M87 in the past, and similarly higher than those reported for all other galaxies. We suggest that most previous ground-based surveys for novae in external galaxies, including M87, miss most faint, fast novae, and almost all slow novae near the centers of galaxies. Comment: 116 pages, 17 figures, submitted to the Astrophysical Journal Supplement Series Minor updates, conclusions unchanged
