All of the times, days of the week and prices mentioned below are likely to rapidly become out of date or otherwise change at short notice. To reconfirm the information, you're best off asking a hotel in Leticia.
By Boat to Brazil - Manaus & Beyond
There's slow boats from Tabatinga in Brazil (Tabatinga is next to Leticia - the two merge together) to Manaus. The boat is far from luxurious,and often very cramped, so try to "reserve / bagsie your prime hammock spot as soon as possible. This is a very interesting, scenic and adventurous way to travel into Brazil - but do bring a book as you'll almost certainly become very bored at stages. Meals are included, but bring plenty of your own bottled water.
The boats depart the Porto Fluvial in Tabatinga on Wednesday and Saturday at approximately 2pm. The journey lasts four nights, costing approximately 150,000 pesos per person if you string up your own hammock, or about 550,000 pesos for two people sharing a cabin room. You can do the same journey in reverse, but it takes longer (about 6 days) as you're travelling upstream in this case. The cost from Manaus to Tabatinga is 250,000 per person in a hammock, and 750,000 pesos for two people in a cabin.
To book this boat to Manaus, go and find the boat at the port a day or two in advance and sort it out locally.
There's no roads connecting Manaus with the rest of Brazil. To travel onwards from Manaus, take a further boat downriver to either Santarem or Belem (which is on the coast) - from both you can start catching buses to various travel destinations in Brazil. Alternatively Manaus has an international airport - there's flights to various destinations.
If all this sounds a nightmare, remember that there's flights from Tabatinga to Manaus with the Brazilian airlines Trip and Rico.
By Boat to Iquitos, Peru
High speed passenger boats depart from Tabatinga for Iquitos in Peru. The journey takes about 12 hours, departing Tabatinga's Porto da Feira at 5am on Wednesday, Friday and Sunday. The journey costs 140,000 pesos in either direction.
Once in Iquitos, you can catch a flight to Lima (try Lan Peru, amongst other airlines), otherwise you'll have to continue by river to Pucallpa, and it will take about 7 days to reach there. Peru's road network begins in Pucallpa.