Kigoma is small and most of the centre can be walked in 20 minutes. For the outlying areas — Kibirizi to the north, Ujiji and the stadium to the south, the airport — daladalas (shared minibuses) run the main routes for a few hundred shillings, and there are plenty of bajaj (auto-rickshaws) for slightly more comfort. Agree the fare before getting in.
For trips along the lake to Gombe and Mahale, you will need to charter a small boat, which any of the lakeside lodges or the park office can arrange. Speedboats to Gombe take about ninety minutes; the slow public lake taxi takes most of a day and is much cheaper but unreliable.