Really Cheap Plane Tickets
To begin with, the best place to buy really cheap plane tickets is online. Travel agencies can't compete with discount ticket sites. Search Google ("cheap airfare south america", for example), or try the well-known sites, like Expedia, Travelocity, Cheap Tickets, Hotwire or Priceline.
Just now I checked fares from Tucson, Arizona to Traverse City, Michigan on the five sites mentioned above. The cheapest quotes ranged from $704 to $432. Nevermind which was cheapest (oh, okay, it was Expedia), because you never know from day to day which sites will find the cheapest flights.
Check at least three places. In this case, it would mean a savings of $272. The flight durations are within minutes of each other, by the way, and the food is probably equally bad on all the flights.
Flexibility For Even Cheaper Airfare
The more flexible you are, the better your chance of getting a really cheap plane tickets. Try different departure and return dates if you can. Check the box "2 or more connecting flights". Mark the "anytime" departure, even if you don't want to leave at midnight - the savings might change your mind. Try all the options. If the savings aren't enough, you can always book first class, direct flights, or whatever you prefer.
Really Cheap Plane Tickets - More Ways
You can buy two tickets. The cheapest airfare from Grand Rapids, Michigan to Tucson, Arizona costs $451 roundtrip right now. Las Vegas to Tucson is only $161, however, and there are regular $199 specials to Vegas from Grand Rapids. It's up to you whether saving $91 is worth booking two flights. (It saved us over $1000 when we went to South America.) There is the problem of whether you'll get a refund if your timing is off on the connections (you probably won't).
Check for package deals, where you pay for plane tickets and your hotel room. I haven't had much luck with these deals, but it can't hurt to look at them.
Buying ahead of time gets you cheaper plane tickets. Any thing less than a week away tends to be expensive, although you never can tell in the mysterious realm of airfare pricing. Sometimes the "last minute specials" are the best thing going.
Priceline still has their "Name Your Own Price" feature. Make your bid, and the airlines say yes or no, depending on whether they have empty seats to fill. First search for the cheapest tickets you can find, then bid lower than that. If your bid is rejected, you can return to the website where you found the best deal and book it.
If You're Really Cheap
My wife and I once took a bus from Michigan to Miami, and flew from there to Ecuador. For us at that time, saving a couple hundred dollars was worth 38 hours on a bus. This won't save you much on domestic flights. Right now, for example, taking a bus from Traverse City to Detroit, then flying to Los Angeles, would only save about $70 (allowing for cab fare to the airport) over flying directly.
There are other options beyond the scope of this article. Courier flights, for example, can be a great deal if you don't mind letting a company use your baggage allowance (you get carry-on only). There are also stand-by deals and last minute deals and other ways to get really cheap plane tickets. Good luck!