We present the first results of a multiyear program to map the orbits of M-dwarf multiples within 25pc. The observations were conducted primarily during 2019-2020 using speckle interferometry at the Southern Astrophysical Research Telescope in Chile, using the High-Resolution Camera mounted on the adaptive optics module (HRCam+SAM). The sample of nearby M dwarfs is drawn from three sources: multiples from the RECONS long-term astrometric monitoring program at the SMARTS 0.9m; known multiples, for which these new observations will enable or improve orbit fits; and candidate multiples flagged by their astrometric fits in Gaia Data Release 2 (DR2). We surveyed 333 of our 338 M dwarfs via 830 speckle observations, detecting companions for 63% of the stars. Most notably, this includes new companions for 76% of the subset selected from Gaia DR2. In all, we report the first direct detections of 97 new stellar companions to the observed M dwarfs. Here we present the properties of those detections, the limits of each nondetection, and five orbits with periods 0.67-29yr already observed as part of this program. Companions detected have projected separations of 0.024-2.0" (0.25-66au) from their primaries and have {Delta}I<~5.0mag. This multiyear campaign will ultimately map complete orbits for nearby M dwarfs with periods up to 3yr, and provide key epochs to stretch orbital determinations for binaries to 30yr.