We present measurements in the context of a survey of southern hemisphere binary and multiple stellar systems observed with the Zorro Speckle dual diffraction-limited optical imaging camera on the 8.1m Gemini South telescope carried out between 2019 and 2023. The overall motivation of our survey and some initial results of these observations are outlined to demonstrate the capabilities and limitations of Zorro. We report on the astrometric characterization of the instrument in terms of the precision and accuracy of our measurements and provide details of our custom-made data reduction pipeline. For targets with separations smaller than 0.4", an overall precision of 1mas in the radial and tangential directions is obtained, while the uncertainty in position angle is 0.2{deg}. Relative astrometry and contrast brightness in the two Zorro filters at 562 and 832nm are reported for 70 pairs on 64 distinct systems (six are triples). Eleven new binaries are found, mostly of small separations (down to 15mas) and large brightness contrast (up to {Delta}m=6 in the red channel). Our results indicate that the Zorro instrument, when properly calibrated, delivers excellent-quality data for visual binary studies of tight and/or faint companions.