Best way to travel through South America?

My wife and I are going to be traveling through South America for a month or so, and I am trying to get a sense of what it is like to fly from country to country. Are flights pretty cheap? Do airlines cover most of the major destinations, or are there large gaps we need to account for? Do any of the countries have low-cost airlines, or are most just major airlines? And do we just buy the tickets when we reach a country, or can we buy them ahead of time before we start our travels?<br><br>I have traveled extensively in Central Asia, so I am familiar with what it is like to get around by bus and taxi, but if there are any tips about the best way to travel there that would also be appreciated! Thanks!<br><br>TLDR: What is the best/cheapest way to travel through South America, and any specific tips?<br><br>EDIT: Thanks for all the advice! 1 month is way too little time for a trip! I keep reading about amazing sites I would not have considered, so will plan for longer.