Buenos Aires is the capital of Argentina, with around 3 million inhabitants.
Buenos Aires is huge, and getting out of it can be a hassle. It is recommended that you get to the edge of the metropolitan area. Buses and trains inside the metropolitan area are cheap.
Any direction. General hitchhiking points
There are places in Buenos Aires where a very big part of the trucks pass by or stop. You could go to the road that goes around Retiro (or Puerto Madero area). On the other side of the park, in front of the train station, there is a crowded intersection. Trucks leave from there to all directions. Or in Mercado Central. Most trucks from there go north, since fruits come from the north.
Southwest/South to Ushuaia (Bahia Blanca, Puerto Madryn, Tierra del Fuego)
Head to the metro station José Maria Moreno and walk 300 metres to the on ramp. It doesn't seem like the greatest place but we had at least a car stop every five minutes. Unfortunately most cars aren't even going until the airport but it's not too far and drivers offer to take you until the airport. There's also a cheap bus (#8) that goes to the airport but you'll have to walk/hitch back (3kms?) to the junction as the driver won't let you out. Once you get yourself to the last freeway junction before the airport, walk down the southern direction 300 metres for a great place to hitch. We were able to get to Ushuaia in four days but is definitely doable in three. Unfortunately the driver ran out of fuel late at night and all the petrol stations were closed and then missed the Tierra del Fuego ferry by five minutes.
Now build new metro station Plaza de los Virreyes, you can climb on the road Carretera del 25 Maio, walk around 500 meters and see tollgate, good place after tollgate. Be careful, you will need to cross higway in one place, for coming to tollgate.
Another option: take a train from station Constitucion to Ezeiza and walk 3km to toll (Peaje). Lots drivers go to intersection near Cañuelas, you can take them, there is not bad place to get a ride futher.
West to Santiago de Chile, (San Luis, Mendoza)
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.
One of the best spots is Lujan paytoll (Peaje Lujan). You can take bus 57 and ask to be dropped at the toll. There are many different routes of Bus nr. 57, and it is convenient that you take the one leaving from Moreno. Once a bus driver told that only the ones leaving from Moreno have official stop at the paytoll because they take the workers there. The same bus driver, on a Bus nr. 57, leaving from Miserere, Capital Federal wanted to refuse to stop at the toll. In the end he stopped anyways.
Another good spot is the YPF petrol station of San Andres de Giles, most of the trucks that are going west stop there because the gasoil is cheaper than the Shell petrol station near Lujan. To go there take the bus 276 from the bus terminal of Lujan and ask the driver to drop you at the YPF petrol station of San Andres de Giles.
You can go to Moreno by train from Estacion Once and then change there to bus 57, route "Rapido". Estacion Once can be reached via the Subte getting off at Estacion Once (Linea H) or Plaza Miserere (Linea A).
Once in Moreno, to find the stop of bus 57, from where the train leaves you, take the bridge over the rails, to the road with many people. There you turn left, walk 1-200 meters, pass by the street which crosses the train lines, and Lujan buses will be facing you.
If you decide to take Bus nr 57 from Capital Federal, you have to get on the one that goes on the motorway "por Acceso Oeste". The route according to GuiaT is: Plaza Italia - Av. Santa Fe - Av. Raul Scalabrini Ortiz - Murillo - Malabia - Luis Viale - Av San Martin - Cruce Av. General Paz - Av. San Martin - Perdriel - Int. A. Campos - Dr. Carillo - Av. J. D. Peron - Av Dr. Ricardo Balbin .... and a lot more. Street names get repeated once you get out of Capital Federal, consider that. Also you can take Bus nr. 57 from Plaza Miserere. If you don't have SUBE card, in most of the stops you have to pay about 20 pesos in coins, or ask someone to pay with their SUBE card for you.
In Both cases you can also decide to not get off at the Lujan toll, but stay on the bus until Basilicum de Lujan, that is also a good spot at the exit of the city Lujan.
For those hitching to Chile, remember that taking food through the border is prohibited.
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 2.7 Pesos (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 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!
This option is for those looking to get to Iguazu. From the main train station Retiro take Linea Suarez. There is a train every 15 minutes on week days, slightly less regularly on week-ends. In Villa Ballester change to the one going to Zarate (you should buy a ticket for this one). You can get off in Campana. Buses to Campana peaje pass on the street in front of Campana railway station. They go around the park and turn right at the corner of it.
This is a long way to get to the hitching point. The train is slow and you have to wait for connection in Villa Ballester. As the train from Villa Ballester to Zarate only runs a few times a day it is not advisable to use this option without checking the timetable of this connecting train first. A more direct way is to take a bus to Zarate, from a small terminal nect to metro station Once. There are three bus options, differing in price and frequency of stops. If you ask nicely,the driver can drop you off at the roundabout before entering Zarate, which is a 5min walk from the toll booth going north.
Other possibility : From the city center of Buenos Aires, take the bus 60 and go out at the terminus in Escobar. It takes around 3 hours but it costs only 5 pesos ! In Escobar, you should go to HIPERMERCADO JUMBO, on the street called Ramal Escobar. Then you will see the shild with the direction of Campana and Zarate. You can start hitchiking. (Le monde sur le pouce, october 2013). You can also use any other local bus that goes to the bus terminal of Escobar and walk a few blocks to the on ramp of the highway from the terminal in Escobar (if in doubt ask people for directions once you get off the bus). From the on ramp of the highway it is quite easy to hitch, there is sufficient traffic and people drive slowly.
Extra information for bus 60. There are 12 variants of the 60. Make sure it says Escobar on the LOWER display panel, and confirm with driver. It also costs about $12 with cash payment. My driver accepted $7 with a persuasive smile. To pay for buses in Buenos Aires you need a SUBE card. If you don't have one you might be able to persuade another passenger to take you on his card and give him back the expense for your trip in cash. Express 60 leaves from plaza Italia, but costs 27$. Ride on normal took 2 hours, but note; it passes the peaje where Ruta 9 peels off (see map) and an excellent YPF station on the highway about 1.5 hours in, both good places. I stayed till Escobar, went to the on ramp and got a ride to Zarate in 10 mins. Demonstrating you are a traveller is a VERY good idea to get out of BA, even if it means hamming it up with a flag or thai fishing pants. Ok, don't go that far. All in, 3 hours to get out of the city, and made it to Montevideo in a long day.
The ferry to Uruguay
The easiest way to get from Buenos Aires to Uruguay is by ferry to Colonia del Sacramento. From the port in Puerto Madero/Retiro leave the main ferries by the companies Seacat Colonia, Colonia Express and Buquebus. Seacat Colonia is part of Buquebus and is usually the cheapest. Booking on their website is the easiest way to get a ticket. Check their website in both the Argentinian and the Uruguayan version. Found out that booking from the Uruguayan site was 50% cheaper: US$20 instead of USD$40 one way. If you don't have a credit card, ask an Argentinian to buy it for you and pay them back in cash.
There's also a ferry leaving from Tigre to Carmelo in Uruguay that apparently goes via Isla Martín Garcia, an Argentinian island that's actually closer to Uruguay in the Rio de la Plata.