In a sense, there is no best, just different. Hitching rides is a Chilean cultural norm; any route offers opportunity. @pnuts is on holiday, I'm a poor substitute, but agree your initial plan of going south and west works.
The most expeditious driving route is along RN 36, making your way out of Córdoba along Av. Velez Sarsfield toward the southwest. It becomes RN 36 outside of the city itself. At Río Cuarto, you go from RN A005, then Ruta 20 in General Pedernera, picking up RN 7 and on to Mendoza. There are lots of little jigs and jogs, but plenty of chances to stop, eat, rest, and interact with the warm and welcoming Chilenos.
For routes on which local buses run from town to town, look at RN 148 or RU 20 and onto 146 (just after El Pantanillo), working your way south to San Luis and to the RU7 and on to Mendoza. Continuing on RU20, although it runs west, may be less desirable: towns are more dispersed, and you'd have to get to RU 142 to make your way toward Mendoza.