South America is a big place and somewhere where you should definitely visit at least once in your life, here are some cities that you should visit, but remember that it is not exhaustive and there are an incredible number of cities you can and should explore there.
Santiago, Chile
Santiago is the beautiful capital of Chile and one of the best cities to visit in South America. Airlines like LAN are constantly flying there from various destinations around the world and it really is an easy place to access.
A great place to start in Santiago would be to see the San Cristobal Hill, which offers a number of fun options from the National Zoo of Chile, to two large public swimming pools, to a stunning view of the city. Other points of interest are the La Moneda Palace and the museum of Pre Colombian Art (Museo Chileno de Arte Precolombino). Just remember that Monday is not museum day in Santiago and most are actually closed on this day!
Bogotá, Colombia
This is the capital of Colombia and truly is the heart of this small country. It is filled with many authentic places to eat, amazing wine and constant food festivals. You can walk through the district of La Candelaria, which is the old quarter, and here in its narrow streets you will find colourful, historic architecture, markets, bars, restaurants and cafes. There is also a lot of graffiti that you can see in the area, that is now becoming more accepted as a cultural part of the city.
Mount Montserrate is another point of interest in the city, it is an iconic hill next to the city and is famous for its stunning view and the church on the top of it, this is one of the best viewpoints of Bogota.
Rio De Janeiro, Brazil
Rio is probably the most famous city in Brazil and is often confused by many as the capital city of Brazil (it is actually Brazilia). Rio is a unique place with endless beaches, samba, a contrasting backdrop of mountains and more samba.
One of the most interesting sights is the Corcovado park which is on top of a mountain that overlooks the whole bay of Rio. There is also the famous Christ the Redeemer statue which sits at the top and is also the largest Art Deco statue in the world, it looks over all of Rio with its arms outstretched.