What a great city!! So much to see and do- we spent 5 days here in total and could have spent 2 weeks! Palermo is the best district to stay in, preferably somewhere near the metro lines. It has great bars, cafes and parks. It's really worth looking into each neighbourhood as well- we did a bike tour, which took in the Recoleto cemetary among other sights. La Boca and San Telmo districts are great too- colourful and characterful!
Insider Tip:Doing the touristy Tango Complejo gives a good overview of the origins and meaning behind the dance. We really wish we'd had to time to go to a 'real' tango experience in San Telmo though.
The Puerto Madryn district has some excellent restaurants, well worth the splurge- we loved Cabana las lilas.
The only place to eat ice cream- Persicco's chain. I can highly recommend the Dulce de Leche casura con brownie!