When I was living in Buenos Aires, Argentina I found that getting around the city was very easy and very cheap! Locals and backpackers travel through the city by the same methods.
Colectivo (Bus)
The bus is the cheapest and easiest means of transportation around Buenos Aires. You will be able to find a bus stop on almost every major street intersection. The buses run 24 hours a day. There are plenty of buses running in the morning through the evening, but when it gets to be past 11:30pm the buses start to be few and far between. You could end up waiting 15-20 minutes for your bus instead of 5 minutes.
When I arrived in Buenos Aires (Bs As) I was so confused on what bus I was supposed to get on to arrive at my desired location. I had never really used the bus in my life, because I always had a car to get around in the States. I quickly learned the only way I was going to learn what bus take was by buying a paperback guide book called "el Guia T." This guide has every bus's scheduled stop and the route that it takes through the city. Without this book I would have spent countless hours riding random buses.
Every bus has a number on the front of it to let you know what route it is driving on. The cost of the bus is always around 80 centavos ($0.25 US Dollars). I say around 80 centavos, because some fares are more than 80 centavos depending on the ride. I would tell you to just pay the 80 centavos, and then nobody will bother you for more money. When you step in the front door of the bus you need to put money into a machine that spits out a ticket. At the same time say "ochenta" (the number 80) to the driver to tell him how much you are paying. Don't ask me why you have to say ochenta to the driver, but all the locals do it and the driver expects you to say it.
For the most part the buses are safe and how most locals get around Buenos Aires.
Subte
The second most popular way for locals and backpackers to travel through Buenos Aires is by the Subte (subway). The subte connects most of the Capital Federal district of Buenos Aires. Starting in the barrio (neighborhood) Congresso through San Telmo, and then all the way out to Palermo. Here is the Map of the Subte in Buenos Aires. During the summer months (Nov. - Feb.) the subte can be extremely hot! I can remember being in Bs As in January, and I was just sweating waiting for the subte.
The subte hours vary depending on which stop you are using. Generally the subte opens at 6 am and runs until 10:30 pm. On Saturday, the subte will run until 11pm and then change back to 10:30pm on Sunday night. On the weekends, the time between subte's are longer than during the week due to less traffic.
The subte is very inexpensive too. A one way ticket costs 90 centavos ($0.30 Us Dollars). When you walk into the subte stop there will be somebody working behind a glass window that you can buy your ticket through. You will be asked for a single fare or a multiple pass (10 fares on one card). It costs the same no matter what option you choose, but it is more convenient to only have to buy it once every ten times.
The subte is safe, but when traveling through Buenos Aires never flash money around and keep your camera put away.
Taxi
There are plenty of taxi's around Buenos Aires. I would generally use a taxi at night when I was going out to eat or to meet up with friends on the other side of town. It is more expensive than the bus and subte, but it is more convenient by being a lot quicker. A fair from Congresso to Palermo is only about 15-20 pesos ($5 or $6 US Dollars). There are meters in each taxi so that you know how much the fare is going to be.
The only tip I have for using the taxi's in Buenos Aires is to be careful when handing them money to pay for the fare. For example, when a driver tells you a fare of 15 pesos and you hand him 20 pesos. Sometimes he will look at you and switch the money in the front seat, and hold it up saying you only gave him 10 pesos. If you can't remember for sure if you gave him a 20 or 10 peso bill then you will have to pay him 5 more pesos. I know it sounds too easy to not fall into this trap, but when you're not paying attention to how much you gave him then they could scam you. It happened to me!