Kigoma is a long way from anywhere. The three options are train, plane and road, in roughly that order of romance.
By train. The Central Line from Dar es Salaam to Kigoma is one of East Africa's classic rail journeys. Trains in theory run several times a week; in practice the timetable is unreliable and the journey takes anything from 38 to 60 hours. First-class cabins (four berths) are comfortable enough; second-class (six berths) is fine for the hardier traveller. Book at the station in Dar es Salaam a few days in advance.
By air. Auric Air, Precision Air and several smaller operators run flights from Dar es Salaam (about three hours, often via Mwanza or Tabora) and direct flights from Arusha. The small airport sits a few kilometres east of town.
By road. It is possible to reach Kigoma by bus from Mwanza, Tabora or further afield, but the roads are rough, the journeys are long, and the buses are crowded. For most travellers, the train or a flight is a much better idea.
By boat. The MV Liemba (when running) connects Kigoma with lakeside villages south to Mpulungu in Zambia. Schedules slip; don't rely on her for a tight itinerary.