Bogotá — a city of contrasts with a hectic balance between the new and the old; the most cultural-minded of South American capitals
Buenos Aires — the city of tango, the most cosmopolitan city of Argentina
Caracas — one of the most cosmopolitan and modern cities in South America, there are lots of theaters, malls, museums, art galleries, parks, well-conserved colonial architecture and even gastronomic restaurants
La Paz — the highest national capital in the world that is built in a canyon
Lima — a curious mix of modernity, large but orderly slum areas, and colonial architecture
Montevideo — the pleasant capital city of Uruguay, situated on the east bank of the Rio de la Plata
Rio de Janeiro — famous for its breathtaking landscape, its laidback beach culture and its annual carnival
Santiago de Chile — capital of Chile with many museums, events, theaters, restaurants, bars and other entertainment and cultural opportunities
São Paulo — a beehive of activity that offers a jovial nightlife and a diverse cultural experience
Getting to South America has gotten much easier in recent years due to massive increases in flights to the continent by major global airlines. Although some particular places are still quite hard to reach (i.e. Paraguay, Suriname, northern Brazil), the places that you most likely want to go, such as Buenos Aires and Rio de Janeiro, are more accessible than ever before.
From Africa: the only (reliable) options worth considering would be the South African Airways service linking Johannesburg with Sao Paulo and Buenos Aires or the Malaysian Airlines service between both major South African cities and Argentina(Johannesburg via Cape Town to Buenos Aires). There are also connections between Luanda and Rio de Janeiro, Salvador de Bahia and Recife with Taag Angola Airlines. Do realize that demand between Africa and South America is very limited, so even the aforementioned services are infrequent and fares may be quite high.
From Asia: Be prepared for a very long journey, especially if your itinerary includes connecting flights to travel to/beyond the major Asian and South American hubs. In addition to the aforementioned Malaysian Airlines flight to Buenos Aires (which does of course originate and terminate in Kuala Lumpur), there is a Japan Airlines service from Tokyo-Narita to Sao Paulo (via New York-JFK, ending in September) and a Korean Air route between Seoul-Incheon and Sao Paulo (via Los Angeles). Do note that the Malaysian Airlines flight makes 2 stops en route, while the latter options involve a stop in the United States that will require all passengers, including those in transit, to pass through U.S. customs. Air China will restart flights between Beijing and São Paulo , via Madrid, in December 2009. If practical, consider the non-stop Emirates flight from Dubai to Sao Paulo, currently the only direct link between the Asian and South American continents. In 2011, Singapore Airlines started flights from Singapore to Sao Paulo via Barcelona
From Australia, New Zealand, and the South Pacific: A somewhat surprising number of options exist. Both Aerolineas Argentinas and LAN Airlines serve Auckland and Sydney from their respective hubs at Buenos Aires and Santiago, while Qantas introduced a non-stop service between Sydney and Santiago on March 2012. LAN also operates one of the world's most obscure flights of all - a service linking Santiago-Easter Island-Tahiti. From Perth Emirates provides affordable flights to Buenos Aires with a stopover in Dubai, around 30 hours total flying time.
From Europe: The entire South American continent once lived under European colonial rule, and the resultant political, social, and economic ties between former colonies and colonizers remain quite strong even today. Portuguese flag carrier TAP Airlines is by far the leading foreign carrier to Brazil, serving a slew of destinations in North and East Brazil as well as the Brazilian capital Brasilia which otherwise have only limited or absolutely no other international connections. Spanish flag carrier Iberia flies to most of the former Spanish colonies, although neither Bolivia nor Paraguay are served. KLM flies between Amsterdam and Suriname and Air France links Paris with French Guiana. Of course, such services are not exclusive - KLM also flies to Lima, TAP to Caracas, Air France to Rio de Janeiro and Lima, etc. Other leading European airlines such as British Airways, Lufthansa, Swiss Airlines, and Alitalia also serve key South American gateways from their respective hubs, while South American airlines also operate into several major European cities as well.
From North America: Until very recently, it was virtually inconceivable to reach South America from anywhere other than Miami. Today, however, rapidly developing hubs at Atlanta, Dallas/Fort Worth, Fort Lauderdale, Houston, Mexico City, New York, Newark, Orlando, Toronto, and Washington-Dulles offer viable alternatives. Indeed, airlines such as Air Canada, Continental and Delta Airlines have successfully begun to challenge the virtual monopoly once enjoyed by American Airlines to several key markets. American discount carriers such as Spirit Airlines and JetBlue (Azul in Brazil) have recently fought hard for and won several route authorities to serve the likes of Colombia, Brazil and Peru, bringing low fares to these markets for the first time. Given the U.S. airlines' cutbacks in service, frequency, and cities served over the years, consider high quality Latin American carriers such as Avianca (to Colombia with daily non-stop flights to Bogotá, Medellín, Cali and Cartagena), Copa Airlines (Panama's national airline to hits hub in Panamá City), LAN Airlines (Chilean carrier serving Chile direct and via several other countries), or TAM (to Brazil).
Beware there are no roads connecting Panama with Colombia, hence it is not possible to drive from Central America. People overcome this problem shipping their cars from Colon (Atlantic side in Panama) to Cartagena or Barranquilla (Colombia), or from Panama City (Pacific side of the Panama canal) to Buenaventura (Colombia) or Guayaquil (Ecuador).
The Pan-American Highway is a series of routes starting in Central America and running to the tip South America. It does not cross through Colombia, but is otherwise an interesting option for those with a good vehicle, plenty of spare parts, and a desire to explore the western edge of the continent.
There are no railroads between Panama and Colombia.
Some cruise liners cover the towns in the lower Caribbean (Cartagena, Santa Marta, Margarita Island). Royal Caribbean, Holland America , Princess Cruises .
The Union of South American Nations gives visa-free access and a customs union between all countries in South America.
There are no cross-country train services in South America, and with the exception of Argentina and Chile, domestic networks are quite limited. There are a number of very scenic "tourist trains" though, including the 445-km Quito-Guayaquil route in Ecuador.
Buses are the main form of land transportation for much of the continent... for longer distances you're often better off flying.
Spanish is the official language in all countries except Brazil, Guyana, Suriname and French Guiana, and is widely spoken even in the countries that are not historically Spanish speaking. Portuguese is the official language in Brazil, which comprises about half the population and land area of the continent. Anyway, Spanish and Portuguese are such close languages that fluency in any of these will allow you to travel within the entire continent. There are also many indigenous tribes living in South America who speak their own languages, and if you are really going off the beaten track, you might have to learn them too. In Guyana, Suriname and French Guiana, the official languages are English, Dutch and French respectively.
Wearing or carrying items which may identify you as an affluent tourist can be a mistake. You shouldn't pack anything that you would be upset to lose. Leave expensive jewelry, watches and other items of value at home and only carry what you need. That goes for credit cards and other documents as well; if you have no need for them leave them behind in the hotel safe, only take what money you are likely to spend with you.
Tap water in many countries is not drinkable, it's wise to purify your own or buy bottled water. Malaria and Yellow fever can be a risk as well on the continent, check with a travel clinic or your doctor before heading out to see if you'll be in a high-risk area.