That sounds like an exciting adventure through some of Japan's most beautiful regions. Here’s a 25-day itinerary that combines public transport and driving, while also keeping you away from the crowds and maintaining moderate pricing. Let’s dive in!

Day 1: Arrival in Osaka

• Morning: Arrive in Osaka. Check into your accommodation.
• Afternoon: Explore the quieter side of Osaka by visiting the Osaka Museum of History.
• Evening: Stroll through the peaceful Nakanoshima Park and enjoy dinner at a local izakaya.

Day 2: Osaka

• Morning: Visit the Sumiyoshi Taisha Shrine, one of Japan's oldest shrines.
• Afternoon: Head to the Osaka Castle Park for a leisurely walk and a visit to the castle (try to go early to avoid crowds).
• Evening: Enjoy a quiet dinner in the Tenma area, known for its local eateries.

Day 3: Osaka to Hiroshima (via train)

• Morning: Take a shinkansen (bullet train) to Hiroshima (about 1.5 hours).
• Afternoon: Visit the Hiroshima Peace Memorial Park and Museum.
• Evening: Try Hiroshima-style okonomiyaki for dinner at a local restaurant.

Day 4: Hiroshima

• Morning: Take a ferry to Miyajima Island. Visit the iconic Itsukushima Shrine.
• Afternoon: Hike up to Mount Misen for stunning views (or take the ropeway).
• Evening: Return to Hiroshima and enjoy a quiet evening.

Day 5: Hiroshima to Shikoku (via ferry)

• Morning: Drive to the ferry terminal and take a ferry to Shikoku (Miyoshi).
• Afternoon: Explore the scenic Iya Valley, known for its vine bridges and beautiful landscapes.
• Evening: Stay in a ryokan (traditional inn) in the area.

Day 6: Shikoku (Iya Valley)

• Morning: Visit the Iya Kazurabashi Bridge and take a short hike in the area.
• Afternoon: Explore the Oboke and Koboke gorges, perhaps take a boat ride.
• Evening: Enjoy a traditional kaiseki dinner at your ryokan.

Day 7: Shikoku (Matsuyama)

• Morning: Drive to Matsuyama (about 2 hours).
• Afternoon: Visit Matsuyama Castle and take a stroll through the surrounding park.
• Evening: Relax in Dogo Onsen, one of Japan's oldest hot springs.

Day 8: Shikoku (Matsuyama)

• Morning: Explore the Shimanami Kaido cycling route (you can rent bikes).
• Afternoon: Visit the 21st Century Museum of Contemporary Art.
• Evening: Dinner at a local seafood restaurant.

Day 9: Shikoku to Kyushu (Kumamoto)

• Morning: Drive to Kumamoto (about 2.5 hours).
• Afternoon: Visit Kumamoto Castle and Suizenji Jojuen Garden.
• Evening: Try Kumamoto ramen for dinner.

Day 10: Kumamoto to Aso

• Morning: Drive to Aso (about 1 hour).
• Afternoon: Explore the Aso caldera and take a hike in the area.
• Evening: Stay at a local guesthouse and enjoy a home-cooked meal.

Day 11: Aso to Beppu

• Morning: Drive to Beppu (about 2 hours).
• Afternoon: Visit the famous "Hells of Beppu" (Jigoku) for unique hot spring experiences.
• Evening: Relax in an onsen and enjoy a casual dinner.

Day 12: Beppu to Yufuin

• Morning: Short drive to Yufuin (about 30 minutes).
• Afternoon: Explore the quaint streets of Yufuin, visit art museums, and enjoy the view of Mount Yufu.
• Evening: Stay in a ryokan and enjoy a traditional dinner.

Day 13: Yufuin to Fukuoka

• Morning: Drive to Fukuoka (about 1.5 hours).
• Afternoon: Visit Ohori Park and the Fukuoka Art Museum.
• Evening: Try Hakata ramen for dinner.

Day 14: Fukuoka to Nagasaki

• Morning: Drive to Nagasaki (about 2 hours).
• Afternoon: Visit the Nagasaki Peace Park and Atomic Bomb Museum.
• Evening: Explore the charming streets of Dejima and have dinner at a local restaurant.

Day 15: Nagasaki

• Morning: Take a cable car to Mount Inasa for panoramic views.
• Afternoon: Visit Glover Garden and the Oura Church.
• Evening: Enjoy Nagasaki's famous champon for dinner.

Day 16: Nagasaki to Kagoshima

• Morning: Drive to Kagoshima (about 3 hours).
• Afternoon: Visit the Sengan-en Garden and enjoy views of Sakurajima.
• Evening: Relax in an onsen and have dinner at a local izakaya.

Day 17: Kagoshima

• Morning: Take a ferry to Sakurajima and explore the island.
• Afternoon: Visit the Sakurajima Visitor Center and hike some trails.
• Evening: Return to Kagoshima for dinner.

Day 18: Kagoshima to Kumamoto

• Morning: Drive back to Kumamoto (about 2 hours).
• Afternoon: Explore any remaining sights in Kumamoto or relax in a local café.
• Evening: Enjoy a quiet dinner.

Day 19: Kumamoto to Oita

• Morning: Drive to Oita (about 1.5 hours).
• Afternoon: Visit the Oita Marine Palace and Aquarium.
• Evening: Enjoy a local seafood dinner.

Day 20: Oita to Kitakyushu

• Morning: Drive to Kitakyushu (about 1.5 hours).
• Afternoon: Visit the Mojiko Retro District and the Kitakyushu Museum of Natural History.
• Evening: Try local specialties for dinner.

Day 21: Kitakyushu to Okayama

• Morning: Drive to Okayama (about 2 hours).
• Afternoon: Visit Korakuen Garden and Okayama Castle.
• Evening: Enjoy a quiet dinner in the area.

Day 22: Okayama to Takamatsu

• Morning: Drive to Takamatsu (about 1 hour).
• Afternoon: Visit Ritsurin Garden, one of Japan's most beautiful gardens.
• Evening: Try Sanuki udon for dinner.

Day 23: Takamatsu to Naoshima

• Morning: Take a ferry to Naoshima Island.
• Afternoon: Explore the art museums and installations on the island.
• Evening: Stay overnight on Naoshima and enjoy a peaceful evening.

Day 24: Naoshima to Osaka

• Morning: Ferry back to Takamatsu and drive to Osaka (about 3 hours).
• Afternoon: Visit the quieter side of Dotonbori for some shopping and snacks.
• Evening: Enjoy a farewell dinner in Osaka.

Day 25: Departure from Osaka

• Morning: Last-minute shopping or sightseeing.
• Afternoon: Head to the airport for your departure.

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This itinerary balances driving and public transport, while also allowing you to explore the quieter, less touristy spots. You’ll get to experience the rich culture, stunning landscapes, and delicious food that these regions have to offer. Enjoy your trip! If you have any specific interests or adjustments, feel free to let me know!