M dwarfs have enormous potential for our understanding of structure and formation on both Galactic and exoplanetary scales through their properties and compositions. However, current atmosphere models have limited ability to reproduce spectral features in stars at the coolest temperatures (Teff<4200K) and to fully exploit the information content of current and upcoming large-scale spectroscopic surveys. Here we present a catalog of spectroscopic temperatures, metallicities, and spectral types for 5875 M dwarfs in the Apache Point Observatory Galactic Evolution Experiment (APOGEE) and Gaia-DR2 surveys using The Cannon (Ness+ 2015, J/ApJ/808/16 ; Casey+ 2016, arXiv:1603.03040; Ho+ 2017, J/ApJ/836/5; Behmard+ 2019ApJ...876...68B): a flexible, data-driven spectral-modeling and parameter-inference framework demonstrated to estimate stellar-parameter labels (Teff, logg, [Fe/H], and detailed abundances) to high precision. Using a training sample of 87 M dwarfs with optically derived labels spanning 2860K<Teff<4130K calibrated with bolometric temperatures, and -0.5<[Fe/H]<0.5dex calibrated with FGK binary metallicities, we train a two-parameter model with predictive accuracy (in cross-validation) to 77K and 0.09dex respectively. We also train a one-dimensional spectral classification model using 51 M dwarfs with Sloan Digital Sky Survey optical spectral types ranging from M0 to M6, to predictive accuracy of 0.7 types. We find Cannon temperatures to be in agreement to within 60 K compared to a subsample of 1702 sources with color-derived temperatures, and Cannon metallicities to be in agreement to within 0.08 dex metallicity compared to a subsample of 15 FGK+M or M+M binaries. Finally, our comparison between Cannon and APOGEE pipeline (ASPCAP DR14) labels finds that ASPCAP is systematically biased toward reporting higher temperatures and lower metallicities for M dwarfs.