There are several buses daily from Dar es Salaam which you can take from Ubungo, Temeke or Mbagala bus stations. Average travel time is about 8 hours. Note that during the rainy season parts of the road can get flooded; it's usually still passable but this can add significantly to the journey time. The companies operating the route change regularly. When picking a bus go for one with four seats in a row, the ones with five can be pretty cramped. Sitting in the back half of the bus can be somewhat traumatic during the non-paved part of the journey.
From Lindi there are buses throughout the morning; in the afternoon you can make the journey by first getting a bus to Mnazi Mmoja Mingoyo then catching an onward bus from there ~2 hours.
From Masasi there are buses throughout the day the last one leave around mid afternoon about 4 or 5 hours.
From Newala there is a bus in the morning, after that there are pickups throughout the day nb. these can be packed and quite painful on the dirt road.
From Mozambique there are pickups that meet the public transport arriving the other side of the border.
Precision Air also fly from Dar es Salaam; their timetable is erratic.
