What is the best route to take when driving from Buenos Aires to Santiago?

Driving Directions in South America.

Take Route 9 from Bs As to Rosario (Santa Fe Province). Then take Route 34 from Rosario to Santiago del Estero Province, go up to LA BANDA city. Then go south (a few miles) to Santiago (Capital city). Distance between Santiago and Buenos Aires is 1150 km (about 720 miles). If you want to take a plane it will cost about Dollars 100.- and the flight takes about 1h40min

If by Santiago you meant "Santiago del Estero", Cecilia up there has hit on the nail. If you meant instead "Santiago de Chile", just take the shortest and classic way from Buenos Aires: Route N潞 7, straight from East to West, cross-country way, up to the Pacific. (Just in case, get info at your hotel or the Autom贸vil Club Argentino about what's still going on with a detour on Route N潞 7 due to Picasa Lake floods).
Once you've left Mendoza city toward Chile you'll be getting right away upslope straight into the Andes. I guarantee you'll never forget the Argentinian landscape on your winding way to the International Tunnel (Los Penitentes, Punta de Vacas, cerro Aconcagua, Puente del Inca).
Starting in Buenos Aires you may also take a touring detour on Route N潞 8 bound Northwest to Southern C贸rdoba province and afterwards take up Route N潞 7 way before getting in Mendoza province; this was advisable a long while ago because of the floods at the aforementioned spot.