Welcome to FlyerTalk!
We will try to help you.
First, are you planning to travel economy, business or first class? What Frequent Flyer programs are you in, and are you Elite on any other them?
Second, have you decided how much time you want to spend in each place? Do you want to go to Rio first and tour after, or vice-versa?
Third, do you want to see other places than Rio and the Amazon? There is a huge variety in a country almost as large as the continental US with four states larger than Texas. There are historical places with 17th and 18th century churches and buildings, mountains, rivers, an Everglades-type of slow river (the Pantanal) that is much larger than the US counterpoint and much more diverse, Iguaçu falls, Bento Gonçalves (our answer to Napa Valley, but smaller and much more rustic, but with some great garage wines to be found) and on and on.
If you choose Copa you will find the prices to be much cheaper than many other choices. For example, I just prices tow business class tickets LAX-GIG-MAO-LAX at $4771.20, which corresponds to the cheapest economy fare I could find of $3082. They use Continental OnePass and give a 25% premium on miles for flights booked on their website. All their flights are B737-800 and their Business class is comparable to US airlines domestic first class but not up to the flat-bed standards of international business on other carriers, but they are much, much cheaper. All, your flights would be 7 hours or less on Copa, some as long as 11 hours on other carriers.