Hi Salsaqueen,
The total travel distance from
Rio de Janeiro to Manaus (Amazon) is 2,939 miles (4,734 Kms). A full table with distances in Brazil can be seen
here.
You should not go to Peru to get to Manaus if you travel from Rio. Work your way to Boa Vista which is just twelve hours or so from Manaus on the tarmacked BR-174 through the stunning tropical forest zone of the Waimiris tribe, with over fifty rickety wooden bridges en route.
This is going to be an awesome trip! Keep us posted!
Best wishes,
Mark