In only 10 days, don't waste time/money by driving on jammed expressways. Take the fastest Shinkansen train b/w Osaka-Nagoya-MIshima.

Pick up a rental car in Mishima and visit Fuji 5-lake in 3 days.

When is your travel timeline ? In summer the chance of viewing Mt Fuji is lower.

To visit Osaka, Nagoya by car is a burden. check out the populations in those areas as compared to SG then you cud imagine how much time/money you might lose due to traffic congestions, lost in navigation, hunting for empty parking lots nearby etc.

What do you want to see at Nagoya in particular. It's less interesting than some other places in Japan.

The true scenic route from Osaka would be going through Kanazawa, Shirakawago, Takayama, Kamikochi (hikding), Matsumoto (Castle) and then heading to Mt. Fuji. You don't need a car for this trip. The only long ride comes at the end when you take the limited express train to Kofu to change to a bus, or the same train to Otsuki to change to another train to Kawaguchiko.

As mentioned above, driving isn't a great idea.

When? Season -- dates -- are all pretty important in planning your trip.

Driving in Osaka can be stressful and wicked! Get out as soon as possible! lol

From Nagoya, you can take the Chuo Expressway. Get on Nakatsugawa Bypass to Rt. 19 to explore Magome-juku. Drive down to Tsumago-juku as well. You could then continue on Rt. 19 to Narai-juku. Then make your way to the Nagano Expressway at Shiojiri. You could go "left" to Matsumoto, or go "right" towards Tokyo.

(If you don't want to go to Magome, Tsumago or Narai, stay on the Chuo from Nagoya and get off at Komagatake. The ropeway is fun. https://chuo-alps.com/lang/en/ )

If you go to Matsumoto, explore the castle and Nakamachi, then get back on the expressway towards Tokyo.

If you got towards Tokyo, get off at Okaya and explore along Lake Suwa. There are 4 Suwa Taisha(shrines). Even if you don't see all four, be sure to see Spring and Fall at KamiSuwa.

https://suwataisha.or.jp/en/

Get back on the Chuo Expressway, going towards Tokyo.

Kobuchizawa on the border of Yamanashi and Nagano is a nice stop. Hoshino Resorts has the Risonare Hotel with an indoor wave pool. It's also famous for horseback riding. Check Canadian Camp.

A fun way to get to the five lakes area is to get off the Chuo at Kofu Minami and follow the signs to Lake Shojiko. 　https://www.mt-fuji.gr.jp/en/access/　　 https://www.mt-fuji.gr.jp/en/　　

From there you can easily get to the other lakes.

Don't know where you are going back to. If you're going back to Osaka, take the Tomei. It's faster. And crazier. Drive safely!

Thank you for to all the advice.

We are going in summer. We are told by friends that the drive is scenic and not stressful, We love the drive in Kyushu, it is scenic and less stressful compared to SG.

Osaka and Nagoya don't use a car. Also avoid mid August when the highways get horribly jammed because of Obon vacation.

When in summer? Mid-August will be a zoo. Avoid August 10-18. Kyushu and Osaka are two different stories. One is fun, the other is maxing out on stress. Streets near the 5 lakes are narrow and windy. Be prepared for tourists to come jumping out onto the road at any time.

Weekdays are best for driving long distances.

>>We are told by friends that the drive is scenic and not stressful

Where did they drive along ?

I'd only recommend driving on the Tomei/Meishin expressways to my nemesis. I avoid driving on those as much as possible.

It's boring, busy with loads and loads of business trucks, highway buses all the way. Due to busy traffic expect an average steering speed of 70-80km/h excluding breaks.

More scenic drives are in rural areas in Fuji 5-lake to the north and Izu Peninsula to the south of Mishima. Another area is the Japanese Alps.

My drive in last Aug.

https://japantravelinsights.com/jp-2023-aug-jp-alps-part-1/

In Kyushu, expressways are comparatively less busy except around Fukuoka.

ps. returned from my 6th drive in Kyushu in end March. hv driven nearly 40 times in Japan from Kyushu to Hokkaido.

Thank you for your advice. Will plan our route accordingly.