The great thing about BA is it's diversity. What neighbourhood will suit you, depends on your preferences (and your budget).
Although you can find milongas and tango schools in every neighbourhood of Buenos Aires, the typical 'tango' barrio's are located in the south of the city: San Telmo, Boedo (and yes, even Once). These neighbourhoods look a bit deteriorated and have some safety issues during nighttime, but especially San Telmo is being gentrified and popular with tourists, while being cheaper than the barrio's i'll mention now.
Palermo is a nice place to enjoy the more avant-garde side of town. Loads of intimate clubs, ethnic restaurants and nice appartments. On top of that there are some huge parks where you can calm down from life in this hectic city. It is very popular with tourists (and expats). Don't be surprised to hear as much English as Spanish in some parts of this neighbourhood!
Down-town (Retiro, San Nicolás, Montserrat) is a place I wouldn't recommend. It's expensive, incredibly busy during daytime and deserted at night.
Recoleta is Buenos Aires at it's best (but very expensive). This is as close to "the paris of the south" as you'll get. It's a safe neighbourhood, there are nice parks, you're very close to the city centre, there are some quality restaurants and good bars. The main arters of the neighbourhood (Av Santa Fe, Callao, Las Heras and Pueyrredón) are loaded with traffic though. It has some tourist attractions such as the Recoleta Cemetary, where you'll also find most of the barrio's nightclubs, restaurants and a big cinema.
In the north you'll find Belgrano. An upscale and more modern neihbourhood. a bit quieter than Recoleta (except for Avenida Cabildo - Excellent for shopping - and Avenida Juramento) but further away from the city centre (20 mins by subte, 30-40 minutes by bus). It's a more residential area.
Villa Crespo and Caballito deserver special mention. These are primarily residential areas, somewhat further from the city centre (about 15-20 minutes by subte), but are located in the heart of the city. Living there is somewhat cheaper than in the upscale neighbourhoods such as Recoleta, Belgrano and Palermo and on top of that you won't be surrounded by tourists (or expats) all the time. Safety isn't really an issue in these neighbourhoods.
There are about 37 (mostly residential) other neighbourhoods in Buenos Aires of which some are worth checking out. But seen as you're new to town, these neighbourhoods are a good starting point.
