We report the detection of variable emission from Sgr A* in almost all wavelength bands (i.e., centimeter, millimeter, submillimeter, near-IR, and X-rays) during a multi-wavelength observing campaign. Three new moderate flares are detected simultaneously in both near-IR and X-ray bands. The ratio of X-ray to near-IR flux in the flares is consistent with inverse Compton scattering of near-IR photons by submillimeter emitting relativistic particles which follow scaling relations obtained from size measurements of Sgr A*. We also find that the flare statistics in near-IR wavelengths is consistent with the probability of flare emission being inversely proportional to the flux. At millimeter wavelengths, the presence of flare emission at 43 GHz (7 mm) using the Very Long Baseline Array with milliarcsecond spatial resolution indicates the first direct evidence that hourly timescale flares are localized within the inner 30 × 70 Schwarzschild radii of Sgr A*. We also show several cross-correlation plots between near-IR, millimeter, and submillimeter light curves that collectively demonstrate the presence of time delays between the peaks of emission up to 5 hr. The evidence for time delays at millimeter and submillimeter wavelengths are consistent with the source of emission initially being optically thick followed by a transition to an optically thin regime. In particular, there is an intriguing correlation between the optically thin near-IR and X-ray flare and optically thick radio flare at 43 GHz that occurred on 2007 April 4. This would be the first evidence of a radio flare emission at 43 GHz delayed with respect to the near-IR and X-ray flare emission. The time delay measurements support the expansion of hot self-absorbed synchrotron plasma blob and weaken the hot spot model of flare emission. In addition, a simultaneous fit to 43 and 84 GHz light curves, using an adiabatic expansion model of hot plasma, appears to support a power law rather than a relativistic Maxwellian distribution of particles.