I've visited South African countries in the past (South Africa, Zimbabwe, Namibia) and was always self driving in national parcs.
How is it about Tanzania? Do they have gravel roads or is all "completely off road"?
Do you have to get any permits to drive into the parcs?
Are there only expensive lodges in/around the parcs or could one get budget accomodation, too? (I don't like camping.)
How about traffic in Tanzania? Is there a lot of traffic in the cities?
The roads in the parks are mostly dirt, but clearly defined and easy to drive in most conditions. (Public buses go through the Serngeti). It's the roads going to the parks that tax your shocks and rattle your teeth. Yes, you need a permit to go into the parks- there are offices at each entrance. The hotels are mostly on the expensive side, and include full board. There is one "cheaper" option available in all of the parks, but of course, that's a relative term. It is so expensive to maintain and build these places,especially in the Serengeti, including having food trucked in, generators for electricity, security, etc., that camping is the only truly inexpensive option. Lonely Planet has a pretty good description of accomodations in the parks. Good luck!
About the traffic, it isnt that bad, except for some of the city where it can be a bit chaotic at times, but nothing special.
I visited Tanzania about 6 weeks ago. I also rented cars in Namibia and South Africa before, but did an organised tour in Tanzania (and kenia). I think possibly this works out cheaper than renting a car (4wd is really recommended, considering road but also to see wildlife), paying the high entrance fees (25-50 dollar...Serengeti more), and hotels (also because hotels/lodges fill pretty quickly).
A good alternative to the very expensive lodges are the serena and sopa lodges, great value but it's not cheap (3 to 4 star lodges, really good rooms, food and people).
The roads in Tanzania were'nt that bad, apart from the road on the Ngorongoro Crater edge and the road between Ngorongoro and Serengeti. In the parks it's not that bad and you drive slowly anyway.
But actually I did not see many tourists with there own cars. Try Leopardtours for a great trip, they were chosen as the number one organisation of Tanzania, and I must say I can imagine.
In only was in Arusha, much more traffic, a little bit chaotic but it has it's own rhythm, nothing problematic.