To give you an idea about the distances in Brazil : The distance from Rio de Janeiro to Manaus Rio- Manaus is a 2 ½ hour direct flight in a jet-plane. One return ticket Rio de Janeiro - Iguaçu costs more than the air-pass! The distance between Rio and Manaus is 4,200 km. From there, several specialized agencies provide tours going inland, but in fact along the river margins. Many of them use boats, which is the natural transport mean of the region.
I entered with keywords "Amazonas" "Manaus" "tour" and found several sites in English which can be reached by mail.
I would take a flight if it was me. Flights can be expensive because effectively they are the only options.
The other alternative ways are not comfortable or safe.
There is no train along Amazon river. During some floods, the margins would be reached and damaged a railway.
Even if you were going on fast motorcycles one way is about 3 days ride - in other words, if you were truly to do this you would be 6 days on the road.
And right now it would be rainy season, so some of the roads would be impassable. Not to mention you'd be traveling through some very undeveloped regions where all you will see for miles and miles are some truck drivers perhaps. It sure would be a cool adventure, but not something to do when you have deadlines to worry about. If you were to go by bus it would perhaps take 4 to 5 days one way, as the buses go more slowly.
What I would recommend is for you to check out many of the other suggestions for side trips from Rio which will take you to beautiful scenery (Paraty, Petropolis, Ilha Grande, Visconde de Mauá, etc) in a 450 km radius around Rio. Then in the future, if you really want to see the Amazon, you should plan for a month-long leisurely trip down the river from Belem do Pará to Manaus, flying back if possible.
Hope my answer has been helfpul!
Try using the "build your trip" flight comparsion tool here:
Flights To Brazil it will
search multiple websites and get you the best price.
Links of interest:
Amazon Jungle Tours Amazon Riverboat Tours Rio De Janiero