Two barrios always on the tourist list are San Telmo & La Boca. They are located right next to each other and can both be explored in a day.
A cute neighborhood of low rise buildings and cobblestone streets, San Telmo is the kind of barrio you picture when you think of Buenos Aires. If you like street fairs, walk along Defensa to Plaza Dorrego on the weekend. San Lorenzo street has some great murals and wall paintings. Note – I’ve heard the area can be a little unsafe at night.
Still very much a working class area, La Boca is popular for its little area of colorful houses around Caminito as well as for the Boca stadium. Situated right by the river, this area was the main entry point into Argentina (hence the name La Boca which literally means the mouth). When poor immigrants first settled in La Boca, they would beg passing ships to give them whatever paints they had leftover, and use these to color their houses in different hues. The area is very touristy but still very picturesque. The 29 bus drops you right at Caminito and picks you up from the same street. Though it is safe to walk around the colored houses, the rest of the neighborhood is still quite poor and I’ve heard that even the short walk from Caminito to the stadium should not be attempted alone. Still, as odd as it feels to see so many tourists concentrated in a small area of an otherwise poor barrio, La Boca is definitely worth a visit.