Buenos Aires has a lot to offer: fancy restaurants, lots of nightclubs, great museums, and so on. But you don't have to go splurge to have fun, here are a few tips to save money in BA.
The following things get you started, be sure to ask locals and walk around to explore even more. The Cementerio de la Recoleta is incredible beautiful and peaceful, be sure to take a look at the grave of Eva Perón. After that a walk through the Jardín botánico de Buenos Aires in Palermo is a great way to continue the day. Be sure to check the websites of museums as they offer free entry on some days, e.g. the Museo Nacional de Bellas Artes. If you love nature, be sure to check out the Reserva Ecologica, which is east of Puerto Madero.
There are a lot of budget options in the barrio of San Telmo. América del Sur Hostel was recommended to me: the staff is super-friendly and helpful, it offers a good price-performance ratio and is located close to the top sights of BA.
San Telmo is the best option in order to eat cheap. Be sure to wander around, there are lots of places which have special deals between 100 and 200 Pesos. If you don't want to sit down, there are a lot of street vendors in some barrios, in La Boca for example you can get excellent Empanadas for around 10 Pesos. If you want to eat and have a beautiful view go east of Puerto Madero, the promenade which separates the Reserva Ecologica to the barrio has several food stands which offer great local food at low prices, e.g. a Choripán starting at 18 Pesos.
Although cabs are really convenient and quite cheap, there are even cheaper ways to get around the city. If you're exploring one barrio at a time, your best bet is to go by foot. The Subte network might be not that extensive, but it takes you to the most important places for only 4,50 Pesos a ride. The bus is even cheaper, only 3,25 Pesos a ride, but the network is very complex, be sure to ask the hotel reception for which bus to take. If you want to go to barrios or cities which are farer away, take the train, the prices are around 2 - 6 Pesos, and it takes you to Belgrano, Quilmes and even until Tigre. For all mentioned opportunities be sure to buy a SUBE card (25 Pesos, available at all kioscos and also valid in other cities in Argentina).