Hello, Brazil is a huge country and Rio and Amazon are in almost oposite extremes. If you must do them both in one trip, its possible, but I think you should go for Rio (at least five days, including the day you arrive) and then someplaces near, like Buzios (if you're into beaches), São Paulo or even some city along the coast, as Salvador, Recife or Natal (more beaches).

If you say your interests, its easier do recommend something.

Thanks phpacha. We like city life, good food, but also appreciate the great outdoors and nature. We are not beach goers. We'd go for a couple of hours and feel bored.

Thinking in city life AND great outdoors, I would definitely recommend Rio. For a first trip to Brazil (its a first trip, right?) with only 10 days, Rio is the best city to get a sense of Brazil. As I said, it a huge country and with great differences from region to region, but I think that Rio is the best place to start to know it.

That said, I think that five days is the minimum to see the best sights, although the ten days could be easily filled. Here's a good (but small) list of these - if you have doubts about they, I'll try to anwser: http://www.lonelyplanet.com/brazil/rio-de-janeiro/travel-tips-and-articles/76505

If, after some research, you think that it's important to see other cities, I think São Paulo is a good contrast to Rio (most city life and fewer great outdoors) and it's very close (only 1 hour by plane).

I would recommend also Foz do Iguaçu, with a city life far less interesting, but with great outdoors (the Iguaçu Falls, but not only) and very much closer to Rio than Amazon.