I lived in Buenos Aires for three months. You should definitely go to Siga La Vaca in Puerto Madero. All you can eat steak restaurant. Everyone got a bottle of wine too with their meal. When I was there I paid about $15 or about 75 Argentine Pesos for all of that. For that matter there's a lot of nice restaurants in Puerto Madero.
Visit the following neighborhoods Palermo, Boca (although go during the day and be with a big group of people), and Recoleta (Evita Peron's grave can be found here).
La Casa Rosada (their White House) and La Plaza de Mayo (I think it's Thursday afternoons at 5PM when the mothers of los desaparecidos do their march in the plaza).
Calle Florida (good for shopping).
I recommend going to Tango dancing clubs. If you're looking for good night clubs go to Kika or Asia de Cuba.
If you're studying abroad and looking for Americans in Buenos Aires, I recommend going to El Alamo (bar). Also go to Colonia, Uruguay if you have the chance. It's an hour ferry right from Buenos Aires.
Also never pay taxi cab drivers with a 100 peso Argentine bill. Sometimes they try to pass you counterfeit money. Also you don't tip the cab drivers either.
¡Espero que la pases re bien, che boludo!