Considering that your initial post just asked about Rio and then you added that you wanted to see the Amazon, it struck me as soeone saying they wanted to know what to see in New York City and then said, oh, yeah, by the way, they'd like to see Yosemite.

In all seriousness, have you considered scanning a guide book? You may want to see the Amazon, but you need to take that into account in calculating air fares (and domestic flights are not inexpensive in Brazil unless you can get a sale, or maybe a Brazil air pass). It's basically a full-day trip, flying, each way (unless you could arrange to fly home from Manaus; TAM flies non-stop to Miami). If your flights to/from Rio are set, you will use up a fair amount of time, and money, getting to Manaus (and then exploring from there, as there's not much of interst in Manaus itself). BTW, although it would be "winter" in Brazil, the Amazon is along the equator, and the seasonal differences are more wet vs. dry than hot vs. cold. June would generally be drier than, say, December, but you can get showers at any time.

Brazil is the size of the continental U.S., so you have lots of choices; not simply Rio-and-Amazon vs. just Rio. There's lots to see in the general area around Rio, including historic colonial towns in Minas Gerais (e.g., Tiradentes and Ouro Preto), and on the coast (Paraty) and Ilha Grande. If time and money permit, you could consider flying to the city of Salvador, or Iguaçu Falls (both are worth seeing, and a lot closer than Manaus). I'm not very familiar with Sao Paulo, but some people like vistiing there, for shopping, restaurants, museums, etc.

Brazil is not a cheap country to visit, but you're spending a lot for your flight, etc., so make the best of it. Hence my suggestion to get a guide book. I reside in the Northeast of the country.

Don't forget your visa and, if you do decide to go to the Amazon (or the Pantanal, which is better for wildlife viewing) anti-malarial prophylaxis.