Having just done this (I won't be in BA until end of the month), my main suggestion would be to rent a furnished apartment for your first month. Once you are there, you can get a better feel for what areas you are comfortable with and also possibly avoid the whole runaround of ridiculous agency fees and whatnot.
We've used http://www.inmogestionrent.com.ar but I also found this company today and they have some really nice places: http://www.waytobaway.com. Having found some apts on each site - the price difference reflects one incorporating the rental fee and the other charging it to the client openly. (Neither is appealing as it's 20% for rentals under months).
But, it's worth it to have a nice place in the area you want for the first month and then you can see where you'd like to be. I know Palermo reasonably well so decided to try living in San Telmo this time and see what it's like to live there.
I wrote this guide to housing last year, it's still relevant except prices have gone up. More specific advice really depends on what your price range is and how long you plan to stay, but start with the guide and then come back with more questions.
I ended up renting a nice furnished place off Craigslist which is more money than I was planning on spending, but like someone else advised, I wanted something temporary until I got to know the city.
Personally, I decided on Recoleta because it's considered safe and it's close to the chess club, which I frequent on a regular basis. However, I could also recommend Palermo which has some very nice streets and overall a good vibe. Of course, Palermo is quite expansive and varies a lot, but generally, Palermo Viejo and Soho are nice areas. If you would prefer something more urban, perhaps Barrio Norte would be a good option too.
Hope this helps, can't speak for the other parts of the city since I don't know them too well. Though I have been to La Boca and Barracas and can tell you I wouldn't feel comfortable living there.
Good luck and PM me if you have any other questions.