Flights to Western DRC including Boma, Matadi and Muanda use the convenient domestic-only N'Dolo Airport; all airlines have ticket offices here. For anything further afield, such as to Eastern and Southern DRC, including Kisangani, Goma, Bukavu and Lubumbashi, you must head out to N'Djili Airport.
Give yourself plenty of time to get to the airport. A pre-ordered taxi from the city centre to N'Djili Airport will take over an hour and cost US$40 at an absolute minimum, though you can probably flag a taxi down on the street and get it for US$30. Going from the airport into town the smart and safe Jeffrey Travels taxis outside the terminal building will ask for US$45, though you can usually get a local taxi for US$35-40 if you're keen to save money. One budget way to get into the city is to take a shared taxi from outside the airport grounds, which will charge CDF1000 to the centre ville (town centre) or Matonge.
There are multiple arrival and departure points for ferries in Kinshasa, but they're all fairly closely located to each other in Gombe. Boats to and from Brazzaville leave from Beach Ngobila, while those heading up the Congo River to Kisangani leave from the nearby Port Public and Gare Fluviale. All three ports are notorious for shakedowns and general unpleasantness, so proceed with care, and if possible go with a local.
There is no bus station in Kinshasa, and few travellers will find themselves using buses given the vast distances involved to other cities in DRC. Buses to Matadi (CDF15,000, seven to eight hours) leave every morning between 7am and 8am from Place Commercial de Limete, south of N'Dolo Airport. If you want to go to Matadi later in the day, or just to nearer points like Kisantu, you need to take a taxi from Rond-Point Nganba, much further south. For Luanda (Angola) head first to Matadi and change there.
From Kinshasa's small train station you can take various services during the week, but the only one known for its reliability is the Saturday train to Matadi (1st/2nd/3rd class ticket US$20/12/8, eight hours), which returns to Kinshasa on Sunday.