What is the best time to go traveling in south ...

The official high season in south america is during the summer- December to March. The only meaning of "high season" is that all the resorts and tourist attractions would be packed... Many parts of south america can be triped in months considered as "low season", for example- during january you can go to Nazca lines in peru. August and september would be a nice time to visit the Galapagos or Machu Pichu. South America is a great destination throughout the year,  It all depends on what kind of trip you would like to have. 

Given the variety of different locations and climate difference throughout South America, I have listed a number of countries with the best seasons for travel. Please be aware that the seasons are reversed in the Southern Hemisphere:

Spring - September to December
Summer - December to March
Fall - March to June
Winter - June to September

Argentina
Buenos Aires - any time
Iguazu Falls - winter or spring
Patagonia - summer
Andes for skiing - winter
High season is January and February

Bolivia
Avoid the rainy season in the lowlands between November to March
Altiplano - winter
Lowlands and rain forest - April to October

Brazil
Cities - any time
Carnaval in February
Amazon - avoid rainy season between November to May
High season is November to April

Chile
Northern Chile - any time, November is best
Middle Chile - September to February
Southern Chile - December to March
Easter Island and Juan Fernandez - March
High season is January - February and July and August for skiing

Colombia
Dry season in December - March in the mountains;

December to April and July to September on the coast
High season is December to February

Ecuador
Rainy season is December to May
Galapagos - January to April is hot and humid; tour boats docked in September and October
Coast - avoid rainy season
Quito and highlands - any time of year
Amazon - avoid rainy season
High season is December to February

Falkland Islands
October to March

French Guiana
Rainy season is January to June, heaviest rain in May
Travel between July to December
High season is February Carnival

Guyana
Best time is mid-October to mid-May, after rainy seasons ending in late January and late August
Travel in the interior is best in dry season

Paraguay
Rainy season in December to April with frequent flooding in the southeast very hot between October and March
Dry season travel (June to August) is easier.

Peru
Lima - avoid foggy time in June - December
Coastal desert - any time of year
Andes - avoid rainy periods between October - April
Rainforest - hot and humid all year; avoid January - April
High season is May to September

Suriname
Rainy season is April to July, December and January
Best times are mid-August to early December

Uruguay
Best time to visit is mid-October to late March; be prepared for heat and humidity
High season is January and February

Venezuela
Dry season is December to April; but May to October is still nice and prices are lower
High season is Christmas, February Carnival and Easter

Well,let's say you should time it to start at the start of spring of North South America and wind your way down south with the summer. Now you want to arrive at the bottom in patagonia for the summer there, or rather the early autumn: Feb-March. So you could start in the North with Equator or Venezuela around October. And then everything else depends on your budget! 

good luck