We present Event Horizon Telescope (EHT) 1.3 mm measurements of the radio source located at the position of the supermassive black hole Sagittarius A* (Sgr A*), collected during the 2017 April 5-11 campaign. The observations were carried out with eight facilities at six locations across the globe. Novel calibration methods are employed to account for Sgr A*'s flux variability. The majority of the 1.3 mm emission arises from horizon scales, where intrinsic structural source variability is detected on timescales of minutes to hours. The effects of interstellar scattering on the image and its variability are found to be subdominant to intrinsic source structure. The calibrated visibility amplitudes, particularly the locations of the visibility minima, are broadly consistent with a blurred ring with a diameter of similar to 50 mu as, as determined in later works in this series. Contemporaneous multiwavelength monitoring of Sgr A* was performed at 22, 43, and 86 GHz and at near-infrared and X-ray wavelengths. Several X-ray flares from Sgr A* are detected by Chandra, one at low significance jointly with Swift on 2017 April 7 and the other at higher significance jointly with NuSTAR on 2017 April 11. The brighter April 11 flare is not observed simultaneously by the EHT but is followed by a significant increase in millimeter flux variability immediately after the X-ray outburst, indicating a likely connection in the emission physics near the event horizon. We compare Sgr A*'s broadband flux during the EHT campaign to its historical spectral energy distribution and find that both the quiescent emission and flare emission are consistent with its long-term behavior.