Driving between Fuji and Takayama?

On our trip to Japan, we want to spend 3 or 4 days exploring the area Mt. Fuji - Takayama - Kiso valley. I thought it might be best to rent a car. I know it's possible with public transportation, but a car would give us much more freedom.
We will be in Tokyo first. What would be a convenient place to rent a car? I thought of first taking the train to Kifu, Matsumoto or Shizuoka and than rent a car there. Any thoughts?
How are the roads in this area? I think we want to take smaller roads as well.
And what about driving times, how long would it take to drive from Fuji 5 lakes to Takayama? And to Kiso valley?
Any tips or ideas are welcome.

Yes, this is a good area to explore by car. Considering all the destinations you plan to visit, I strongly recommend to allocate at least four days to your trip.

What would be a convenient place to rent a car?

I assume you will return to Tokyo afterwards. In that case, renting a car in Tokyo would be most convenient. If you wish to avoid driving in central Tokyo, you could chose an outlet in the suburbs, such as Hachioji.

How are the roads in this area?

The major roads are well maintained and relatively wide. There are also expressways between the Fuji region and the Kiso Valley.

And what about driving times, how long would it take to drive from Fuji 5 lakes to Takayama?

First about two hours from Kawaguchiko to Matsumoto along the Chuo Expressway and then another 2 hours over the mountains to Takayama.

And to Kiso valley?

Takayama to Kiso Valley takes about two hours. And from there about 3.5 hours to return to Tokyo.

Thank you, this is very useful information.
Actually we will probably not return to Tokyo but head for Kyoto instead. We will probably have a railpass, so that's why I thought it would be convenient to take the train to somewhere like Kofu first, before getting a car.