<map lat='-34.6' lng='-58.45' zoom='11' float='right' width='400' height='340' view='3' /> Buenos Aires is the capital of Argentina, with around 12 million inhabitants. It is probably not easy to hitchhike here.
West to Santiago de Chile (San Juis, Mendoza)
Buenos Aires is huge, and getting out of it can be a hassle. This method is not necessarily the easiest, and you could almost certainly hitch closer to the city.
Trenes de Buenos Aires runs some of the commuter rail lines in Buenos Aires, including Linea Sarmiento from Estacion Once. Estacion Once can be reached via the Subte from Stations Once (Linea H) and Plaza Miserere (Linea A). From there, you'll want to catch a train going to Lujan. Depending on the time and day, you may have to transfer at Moreno, though it is a seamless transfer and one that many others will be making.
Once you arrive in Lujan, you have a couple of choices. You could walk the several km to to highway, but that might not be your bag. There should be a bus waiting outside the station with a sign for the Basilica. Surely the most famous thing in Lujan, the Basilica towers above the city and region, and is truly quite striking. The bus is only a few pesos, and will take you right to the Bascilica, which is a very short walk north from Ruta 7.
Ruta 7 is the primary highway (though still very small considering that) east-west across Argentina from Buenos Aires to Santiago de Chile, and everywhere in between. Once you get to Ruta 7, there is quite a bit of shoulder (though know that the traffic does not pick up the entire length of the highway!)
East to Mar del Plata trough ruta 2 and the coast
From wherever you are in Buenos Aires you need to get to Constitucion, end of the blue line of the south of Buenos Aires. It will cost you the normal city fare of 1.1 Peso (20 eurocents) and from there it is recommended to take a train direction to La Plata. You will get out on train Hudson.  Then do not go to the closest exit to the highway - Xsuzann has been caught over there by police and sent away. Cross the railway and head to the road which seems to be a road going to the country - but its actually an exit for highway. You would have to walk about 2 km, but it's worth that, as the cars don't go that fast on the exit and it's very likely that someone will take you!
From the main tren station at Retiro, which you'll find in the center of Capital, there are trains that depart twice daily for the city of Zarate, which is well outside of the city. It costs 3.5 pesos for the two and a half hour train. There are quicker, more expensive, and more exhausting ways of getting out of town, but this one is a clincher for those of you who aren't in a rush. As of this writing, the earliest train departs at 14:00 and arrives to Zarate at 16:17. It isn't ideal for a good-timed hitch, but it's the most straight forward and cheapest.