Arusha and Kigoma are 800 km from each other, as the crow flies, but by road, Kigoma is about 1,140 km from Arusha: no short travelling distance.
The easiest way to get from Arusha to Kigoma is by air, though this is not the cheapest way.
There are no direct flights between Arusha and Kigoma, but you can fly Air Tanzania between these two cities. This journey with Air Tanzania will include stops in both Zanzibar and Dar es Salaam.
If you want to travel by road or train from Arusha to Kigoma, you’d most probably have to catch a bus or train from Arusha to Dar es Salaam, and then have a second leg of this trip by bus or train from Dar es Salaam to Kigoma.