From the Airport
A cab is the best way to get into the city; it's inexpensive and convenient.
Just before the exit from the arrivals hall is a currency exchange booth. It isn't necessary to have pesos to pay for a cab, but if you want to tip in pesos, it's a good idea.
When you exit the airport, there are booths on the sidewalk where you can pre-pay the cost for the cab (approximately $18). You can use a credit card. When you pre-pay, you are given a receipt showing payment, which you present to the driver. These are licensed "Radio Taxis," and are safe. It is not a good idea to select a cab by yourself when you exit the airport.
Other options include "Remises" (private drivers), rental cars if you're going to have one while in Buenos Aires, or the shuttle bus, which is substantially less expensive ($8) and a lot more time-consuming. You can get the shuttle bus at the booth labeled Manuel Tienda Leon; it brings you to a central terminal near the waterfront and you will have to get a cab there for arrival at your hotel, or change to a "Collectivo," another bus service. Note, too, that this is a per-person price, while a cab can take four people -- making a cab substantially more cost-effective and convenient.
From the Port
Again, unless you have a private driver or arrangements through your cruiseline, a cab is definitely the best option. Look for the booth located just outside the cruise terminal.
From the City to the Port or Airport
Have your hotel concierge or front desk receptionist contact a Radio Taxi or Remise for you. Do not take a cab on the street; it might be fine but you run the risk of being grossly overcharged. Remises are like limos, usually better cars with well-dressed drivers ... figure on $35 plus tip for a trip to the airport and $18 plus tip for a trip to the port.
Buenos Aires has an ingenious bus system called "Collectivos." They are small buses, not quite van-sized, but smaller than regular buses. The cost is about .27 cents per trip (80 Aregentine centavos ... have coins because they don't take bills.) They run 24/7 on 144 different routes! You can get maps from your hotel concierge or from any tourist booth in the city or at the airport.
The "Subte" is an extensive network of underground transportation. Again, you can get maps at any tourist booth or from your hotel concierge, or you can print one out here. There are five lines, which go in all directions, and what a bargain they are at about .25 cents (70 centavos) a ride!
This is the best way to travel long distances, travel with luggage, and travel with tired children. Even if you are in a restaurant, someone will call for you. If you aren't going to the airport or anywhere long distance where there is a possibility that you will be "ripped off," you can flag one down on the street. Make sure they are regulated Radio Taxis: a black car with yellow roof. A free cab will have their sign lit up with the word "LIBRE." The meter charges $2.00 initially and .25 cents every 200 meters.
If you want to venture outside of the capital city -- to Tigre, for example -- there is a network of trains with four main stations in Buenos Aires. Most visitors find that the routes out of the Retiro Station are the ones that suit them best, but there are other terminals as well: Constitucion, Once and Federico Lacroze.