Choosing just 5 is hard but I can try.
1. Cementerio de Recoleta – Don’t miss it because you think it’s just a cemetery. If you can’t get enough of cemeteries, take the metro to Cementerio de Chacarita, a.k.a Cementario Viejo. Many times larger than Recoleta, Cementario Viejo (and the adjacent German and English cemeteries) is landmark to the huge waves of primarily European immigration to Argentina.
2. Take a Stroll Down calle Defensa in San Telmo – From cross street Chile to Plaza Dorego you’ll find fascinating antique, art and clothing stores as well as excellent dining. Sunday is the market day. The neighborhood is generally quiet during the rest of the week but can be quiet active in the evening with the various bars and restaurants in the area.
3. Explore Palermo –I’m cheating as the Barrios of Palermo offer a myriad of things to do from shopping in Palermo Hollywood to taking tea at the Jardín Japonés. Palermo offers a lot so take the time to explore.
4. Visit Tigre– Take a break from the city sprawl and spend an afternoon in the Tigre Delta. An hour’s train ride from downtown Retiro Station. From Tigre you can take a lancha colectiva (bus boat) inside the delta wherethe houses are built on stilts. There are several places to stop and have lunch and walk around.
5. Go to La Boca– Specifically, head to Calle Caminito. Sure, it´s very touristic but don’t let that stop you. While the walking street of Caminito is generally safe, to this day the surrounding neighborhood can still be rough, so I don’t recommend wandering too far off.
For more info any guide book can help,