The best route out of Kigoma to Rwanda (Kigali) is via Bujumbura, there are all the way through minibuses at the bus station at Kigoma. The road to the border at Munirra is perfectly tarmacced and takes about 1hr 20 minutes. It costs $20 for a visa to enter Burundi, then there is a rough graded road for 23 Km and one arrives at a small village where the minibuses are swapped, also one can change money at the jungle here! The minibus now RHD continues to Bujumbura and takes 4 hours.
Entering Rwanda for a Brit – there is no visa and no charge. Every other nationality that I observed was turned back at the border and told to go back to Bujumbura and get a visa for Rwanda.