There's no train connection with Buenos Aires Ezeiza Airport. The nearest train station is Ezeiza, and can be reached with bus line 518.
Car and bus
The airport is situated 35 kilometers southwest of Buenos Aires. The airport is easily accessible by rental car: from the airport the toll road Autopista Riccheri (road number A002) leads to Buenos Aires. It's a little more complicated from the city center. From the Plaza San Martín you drive via the Calle San Martín, Avenida Del Libertador, Avenida Maipú, Calle Arenales, Avenida 9 de Julio and the Avenida Dellepiane to the toll road Riccheri. Fortunately the way to the airport is signposted at many locations in the city.
Bus company Manuel Tienda Leon maintains a connection with their Terminal Madero on Plaza San Martin in the center of Buenos Aires. Some busses subsequently continue to Jorge Newbery Airport.
Bus 8 runs from the airport to Plaza de Mayo, but takes a very long time to reach this end destination.
Car rental Buenos Aires Ezeiza Airport
You can rent a car in the Arrivals Hall of Terminal A. Two car rental companies are present at Buenos Aires Ezeiza Airport: Avis and Hertz. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Ticket counters of Taxi Ezeiza can be found in the Arrivals Hall of Terminal A and outside the Arrivals Hall of Terminal B. You pay a fixed rate (approximately 150 pesos). This is by the way not the cheapest rate, the 'independent' taxi drivers outside the terminal charge less. Traveling to Buenos Aires takes about 35-45 minutes.
Terminal A is connected with a large parking garage. Outside lies an uncovered parking lot. You pay 13 pesos per hour, or 78 pesos per day.