  We have conducted a long-term, wide-field, high-cadence photometric
monitoring survey of ~50,000 stars in the Lagoon Nebula \ion{H}{2} region. This
first paper presents rotation periods for 290 low-mass stars in NGC 6530, the
young cluster illuminating the nebula, and for which we assemble a catalog of
infrared and spectroscopic disk indicators, estimated masses and ages, and
X-ray luminosities. The distribution of rotation periods we measure is broadly
uniform for 0.5 < P < 10 d; the short-period cutoff corresponds to breakup. We
observe no obvious bimodality in the period distribution, but we do find that
stars with disk signatures rotate more slowly on average. The stars' X-ray
luminosities are roughly flat with rotation period, at the saturation level
($\log L_X / L_{\rm bol} \approx -3.3$). However, we find a significant
positive correlation between $L_X / L_{\rm bol}$ and co-rotation radius,
suggesting that the observed X-ray luminosities are regulated by centrifugal
stripping of the stellar coronae. The period-mass relationship in NGC 6530 is
broadly similar to that of the Orion Nebula Cluster (ONC), but the slope of the
relationship among the slowest rotators differs from that in the ONC and other
young clusters. We show that the slope of the period-mass relationship for the
slowest rotators can be used as a proxy for the age of a young cluster, and we
argue that NGC 6530 may be slightly younger than the ONC, making it a
particularly important touchstone for models of angular momentum evolution in
young, low-mass stars.
