The less backtracking you do and wasting time, the better and more efficient your travel will be. So starting at Kagoshima and working your way northwards like you suggested would be a good plan. It depends also on where you are coming from before then. If your main international flights are to Tokyo or KIX, then something like the ANA Experience Japan Fare might be one cheap way to get down there, or connecting domestic flights are often cheap add-ons as part of your international ticket.
http://ana.ms/20Xtf8tThe LCCs mentioned above are another good option, although their cheaper fares often require more flexibility in your schedule.
It would sure help also if you gave more info on your interests.
For Aso, if you are up to driving, a rental car is an excellent way to get around. There are some good regional places like the Daikanbo Lookout and Kuju Flower Park that are very worthwhile but really need a car.
Unless you really have your heart set on Yufuin, you might consider Beppu instead. From Hakata it takes as little as 90 minutes and there is overall more to see and do, like the Hells Tour, nearby Takasakiyama Money Park/Aquarium, and other sights.
http://bit.ly/2pZ9opzWith Kumamoto Castle severely damaged there is a lot less reason to spend a lot of time there. The Suizenji Garden is still there and very nice though.
http://bit.ly/2HopvoCNagasaki is the jewel of Kyushu though and its wide variety of history and cultures is only matched by its great food - be sure to try the chanpon and saraudon for lunch or dinner while there.
http://bit.ly/2ob4u8iSeeing Fukuoka is well worth some time as well. The Nanzoin Temple is one of Japan's finest, yet unknown to most. There are some very nice traditional gardens as well, like Rakusuien, Yusentei and at Ohori Park. Some people keep repeating there is not much in Fukuoka, but that's mostly because they just don't know what is there.
http://www.thejapanfaq.com/fukpic.htmlhttp://bit.ly/1KAm8cUYou also mentioned sights along the way to Hiroshima - there are some very good ones also. But I recommend sticking close to the Sanyo side and the bullet train line - the Sanin side can be extremely slow and inconvenient to tour. The Akiyoshido Cave is Japan's finest, and walking through huge caverns and an underground river rushing past you in something you won't forget. Iwakuni is also a very nice stop for its bridge and castle. Plus nobody has mentioned Miyajima which is one of Japan's most famous places.
http://bit.ly/2CgIvaRhttps://goo.gl/fAz4axIf you like, there is also the Yamato museum in Kure, as well as Okunoshima (Bunny Island) where you can get mobbed by rabbits.