I went to Buenos Aires in June 07. One of the best cities I have ever been to.
There is something to do in Buenos Aires every day and night of the week.
The flea market in San Telmo is great, they run on Sundays
If you are in Argentina make sure you see some Tango. There are plenty of restaurants where you can eat and watch at the same time.
The best district to stay is Palermo near the Plaza Italia metro stop.
There is a great place called Tierra Santa http://www.tierrasanta-bsas.com.ar/ - a theme park based on the bible. All made out of fibreglass!
I will get back to you with more stuff when I can review all my tickets etc. almost 10 years ago
You definitely need to stay in the downtown area because you can walk everywhere. On Thursdays the mothers of the disappeared still do their walk in the afternoons, and this is an amazing political (safe) thing to see. They march with their signs of the children they lost from the Dirty War and it is in front of the Pink House. Around there you can see churches and old buildings. Definitely see the Tango show, especially in the local halls for true Tango. The touristic shows do a lot of acrobats and its not as sensual. almost 10 years ago
If you like soccer you should go to the La Bombonera stadium, where Boca Juniors play their games. It is an amazing stadium and their supporters are really fanatics. The La Boca neiborhood is also a great place to see.
I would suggest you the Milhouse hostel. Great place to stay, lots of people and a lot of fun. One of the best backpacks i had ever been. almost 10 years ago