Giorgio's House Buenos Aires Bed and Breakfast has excellent rates. A double private room runs $22 with a minimum three night stay. Each night after the fifth night will cost $19.50. There are also other perks to staying with Giorgio House. When you stay three nights, you get a free Tango (the most popular nightlife activity in Argentina) lesson. When you stay five nights, you get a free Spanish lesson.

http://buenosaires.giorgioshouse.com/
The San Telmo neighborhood is a popular area to visit. On the weekend Defensa Street closes and pedestrians take over. On Sunday there is the popular Denfensa Street fair and market. It is full of antiques, arts and crafts, salsa and tango performers. The old cobblestone roads also make it interesting.
For cheap eats at the market try Pizzeria Parillo's, a slice of cheese filled pizza will cost around $1. A choripan is one of the local favorites. It is chorizo sausage on a baguette bread. Try it at Carlos Calvo and the cost there is about $1.

Choripan: A favorite Argentina street food.
Tango dancing is one thing to experience in Buenos Aires. You don't have go to expensive tango halls to experience the dance. Hang around the Denfensa Street market on Sunday for a free milonga (tango dance). Local outstanding tango dancers come out to wrap up their weekend with a dance. Cochabamba 444 is a popular club that offers Tango lessons for about $3.
Many of the things that tourists find attractive about Buenos Aires does not have to be expensive. For more information refer to the Lonely Planet. http://www.lonelyplanet.com/argentina/buenos-aires