I was lucky during one of these wanderings to go by road. The air-route from Kigoma to Moshi goes through Dar es Salaam, takes the whole day, and is pretty boring. The road-route, on the other hand, takes two days, passes through five regions, and is rather uncomfortable. But it's fascinating.
There are a few options for you and your car. All start by heading up north, through Kibondo district and up to Kagera before heading east. From there you could, I suppose, head up to Mwanza, pass through the Serengeti, to get to Moshi. Or you could drive south to Dodoma and cut straight east to Chalinze (near Dar es Salaam), and then all the way up north to Moshi. Or you could do what we did, which is cut a diagonal through Singida and Manyara regions, up north east to Moshi.
Along the road we saw dramatic changes in houses, clothing, landscape, crops, industry, development, livestock, the price of gasoline, and concentration of traffic police. At the end of the trip I was left wanting to retrace my steps, but more slowly. That's something I've learned here; to learn more you need to slow down.