Palermo (botanico) for sure.
It also depends on who you are and how long you will be in BA.
Usually expats settle down in Recoleta or good part of Palermo first. After 1 year or two, they move tp other part like Villa Crepso, Palermo Viejo.
Then they get comfortable, marry a local guy/girl, they move to San Telmo, Amagro, Caballito .. even to outside city ..
For beginner, Palermo seems to be your place. Botanic is a lovely section. Palermo Chico is great but maybe too pretentious, if you do not run for president.