Ever since Juan de Garay came ashore from his ship in 1580, founding Buenos Aires for the second time, the area that is now Plaza de Mayo has been Buenos Aires' symbolic and political heart. The city's key institutions – political (the Casa Rosada), economic and religious – remain in the same location to this day; naturally you'll find some of the city's best museums and important historical sights in this area, with further key political buildings lying on the other side of super wide Av 9 de Julio.
To the south, the neighborhoods of San Telmo and La Boca have color and grit as well as some cutting-edge art galleries and atmospheric streets and plazas. The docks of La Boca were abandoned in favor of Puerto Madero, a relatively new neighborhood with a couple of museums and a nature reserve nearby.
The flavor changes as you go north to Retiro and Recoleta, past the vast mansions built by Argentina's elite at the peak of the country's wealth, to the equally elaborate tombs in Recoleta Cemetery they commissioned to house them in death. Continue north and you'll reach Palermo, where the main sights surround the neighborhood's green parks.