In a multi-site photometric campaign to monitor T Tauri stars (TTS) in the Taurus-Auriga cloud over more than two months - dubbed Coordinated Observations of Young ObjecTs from Earthbound Sites (COYOTES), we find all 24 of our target stars show evidence for periodic light variations with periods between 1.2 and 24.0 days. This more than doubles the number of periods published for Tan-Aur TTS. The variations of 20 of these stars can be interpreted as rotational modulation of the stellar flux by surface spots. The periods of the four remaining stars (RY Tan, BD+24°676, TAP 26, and LkCa-21) may correspond to orbital periods of binary systems. Models of the UBVRI light curves lead to the properties of the spots, both hotter and cooler than the photo spheric temperature, which appear to be the principal source of the photometric variability of TTS on timescales of days and weeks. The longest rotational period we measure (12 d, for GM Aur) leads to an equatorial velocity of 8km s-1, which confirms the paucity of extremely slow rotators (Veq ≤ ≤ 10km s-1) among TTS younger than 5 106 yrs. Combining our 20 rotational periods with those published for 17 other Tan-Aur TTS, we find that the Weak-line TTS (EW(Hα) < 10Å) rotate faster than Classical TTS (EW(Hα) ≥ 10Å) at the 99.9% confidence level (according to a K-S test). The mean rotational period for the 11 WTTS is 4.1±1.7d; for the l5 CTTS, 7.6±2.1d. We interpret this difference as evidence that WTTS spin-up as they contract on their convective tracks, while CTTS are prevented from doing so by either (a) their strong winds carrying away excess angular momentum and/or (b) a magnetic coupling between the stars and their inner accretion disks, as suggested by recent models. We discuss the implications of this interpretation for the subsequent evolution of TTS toward the main sequence. In particular, we propose that the different rotational histories of WTTS and CTTS on their convective tracks may account for the large range of rotational velocities observed among low-mass dwarfs in young clusters.