Slow boats to Manaus (hammock/cabin R$400/1620; seven days) leave from the Porto Fluvial every Wednesday and Saturday, plus some Tuesdays, between 8am and 2pm. Arrive early to stake out good hammock space, as boats can be quite crowded.
AJATO speedboats leave Tabatinga for Manaus (R$600, 36 hours) on Tuesday, Thursday, Saturday, and Sunday mornings.
To get to Tefé, the Sunday and Thursday speedboats to Manaus now make a stop in Tefé proper. Otherwise, take a Manaus-bound slow boat as far as Alvarães (R$165, 44 to 48 hours), where you can catch a shared taxi to the village of Nogueira (R$12, 20 minutes) and then a motorboat (R$12, 20 minutes) into Tefé; note that taxis and local boats only operate during the day time.
Bear in mind that departure days and times change regularly; always check in advance.
To & From Colombia
The international border is marked by nothing more than a few moneychangers on the Brazilian side and a Colombian police officer directing traffic on the other side. You are free to move between Tabatinga and Leticia as much as you like, but if you plan to travel onward, even to Amacayacu National Park, you should clear immigration for both countries before leaving town. There's a Colombian Consulate in Tabatinga.
To & From Peru
High-speed passenger boats between Tabatinga and Iquitos (Peru) are operated by Transtur and Transportes Golfinho. Boats depart Tabatinga’s Porto da Feira daily around 5am, Brazilian time (boarding begins an hour earlier) arriving in Iquitos about 10 hours later; fares are US$75 each way. Be sure to get a Brazilian exit stamp the day before; you’ll stop first at the island community of Santa Rosa, where the Policía Internacional Peruviano (PIP) handles Peruvian border control. If you're staying in Leticia, be aware that nighttime taxi rates are triple the daytime rate. Be warned, too, that there are slower, cheaper boats to Iquitos, but they are not comfortable and some are unsafe.
If you just want to get to Santa Rosa, small motorboats go back and forth frequently from Porto da Feria (R$4, five minutes) from around 6am to 6pm.
There's a Peruvian consulate in Leticia, Colombia.