When talking to Argentinians about Córdoba, those who are not from the province usually refer to the city, while the people from there refer to the whole province and not just the city.
Like usually with big cities it is probably a good idea to get a bus to somewhere else. When going south east, you might try to take a bus to Carloz Paz, but the highway straight south has probably more traffic.
Going West (Carlos Paz, Mendoza, San Juan)
As said, Cordoba is too big to hitchhike directly from. Take a bus from the smaller terminal de omnibus, located between Av. Ituzaingo and Av. Buenos Aires on Bv. Arturo Illia. The buses here go into the Sierras to various villages and tourist spots. To go west you need to pick a location on Ruta 20, for example Villa Carlos Paz. I chose Mina Clavero and spent a couple of days relaxing in pools and cascades of the river; after all traveling is about enjoyment. From here Ruta 20 passes a number of small towns and villages in this valley so you are never far from the necessities of life but also never overcrowded like the city- beyond Villa Dolores it becomes quite desolate. The Traffic along this straight is plentiful: I was lucky enough to be picked up within 5 minutes and taken directly to Mendoza.
Another option to go west is by taking city bus (colectivo) number 70 from 27 de Abril street with Mariano Moreno street on the west side of the La Cañada river. Ride it till the the giant junction at Don Bosco, where there's a Shell petrol station opposite the bus stop. The route name is E55 and leads past La Calera to the San Roque artificial lake and Carlos Paz. From there you can either choose to explore the Sierras Chicas and/or Grandes de Córdoba as done by Mind of a Hitchhiker, or hitch on to Mendoza via Villa Dolores.
Going East (Rosario, Buenos Aires, Uruguay)
From the bus terminal in Cordoba, catch a bus to Toledo. They sold tickets at desk 18 for 39 Argentinian Pesos (January 2018). Ask locals or the bus driver to let you out at the first (probably only one) bus stop in Toledo, otherwise you might miss the village. The bus ride takes about 40 minutes. At the bus stop in Toledo, there's a small petrol station on the other side. Next to it (left), there's a road going through town. Walk along this road for approximately 500m until the end of town. There, you'll see the highway in the far distance lingering behind the endless fields of the pampa. You either need to walk about 3 km to the highway or be lucky and hitch a car that might go there. Once at the on-ramp, you can walk another 500m along the highway and you will be located directy at a toll station, which is a perfect spot for catching all traffic Rosario and/or Buenos Aires bound. To be fast, make sure you'll only hitch a car that goes minimum to the next toll station. Petrol stations along the way are rare and not worth it. Go for it, toll stations!