Iquitos is a city in Peru. It is the largest city in the world that is not accessible by cars.
From Iquitos, you can go to Nauta by car, using the highway. Just take a bus or walk to go out of the city and waive your thumb. It is approximately 2 hours to go to Nauta
Slow boats out of Iquitos
You have basically 4 direction out of Iquitos. All these boats includes food. You have to bring your hammock (you can buy some for 15-20 soles. Possibility to pay more to have a cabin.
1) To Pantoja at the border of Ecuador. Once or twice a month. 80-100 sols (~33$). Between 5-7 days. Stop at Indiana, Mazan, Santa Clotilde
2) To Santa Rosa at the border of Colombia. Every day from Monday to Saturday. 70 Sols (~26$). 2-3 days. Stop at Pevas, Chimbote and Caballococha
3) To Yurimaguas in Peru. 3-4 times a week. 100 Sols (~33$). 2-3 days. Stop at Nauta and Lagunas
4) To Pucallpa in Peru. Monday, Wednesday and Friday. 120 Sols (~33$). 4-5 days. Stop at Requena and Contamana
Hitchhiking in or out by the river
It is maybe possible to hitchhike a boat, but the distance are quite far. Only the regular slow boats are going all the way on big distances. You have also quite a lot of cargo ship. It is worth a try to see if it's doable.