Hi. To get to the southern paradise of Tokushima, you may consider these 4 options. First, Cross the Great Akashi Strait Bridge by highway bus. There are mainly three areas in Kansai from where the highway buses depart, Osaka, Kobe and Kyoto. Tickets can be reserved and bought online, but unfortunately only Japanese is available on the website, so in case you don't read/speak Japanese I'd recommend making a reservation by visiting any bus terminal/travel agency directly and buying your tickets using your formidable international body language. Second, Cross the Great Seto Bridge by train. As this is quite a detour and rather expensive, I wouldn't recommend this route. However, if you want to do it by train, then this way becomes amenable. Third, Cross by ferry boat from Wakayama. To start with, you have to get to Wakayama Port by train. Fortunately, there's a direct train available from Nankai Namba Station. Take a Nankai Express Southern train bound for Wakayama Port, which would stop at Kansai International Airport. It takes about an hour, 920 yen one way. Then take a Nankai Ferry Boat bound for Tokushima Port, which takes about 2 hours, 2,000 yen one way. In case you take the ferry boat along with your car, the fee for the car would be 5,800 yen ( if vehicle length is shorter than 3 meter ), 7,600 yen (if longer than 3, shorter than 4 meter), etc. Lastly, By car. If you care for driving and can avail of a car rental service, of course you can drive to Tokushima! It will take about 3 to 4 hours depending on how fast you drive or how heavy the traffic is.
Onto your next query, the cheapest and quickest way to arrive at Kagoshina is via plane from Osaka Kansai Airport. There are no direct bus/train from Osaka Kansai Airport down to Kagoshima. But the best way to get from Osaka Kansai Airport (KIX) to Kagoshima without a car is to take a train which takes 5h 22m. If you prefer to leave by bus, Osaka Kansai Airport (KIX) to Kagoshima bus services, operated by Miyazaki Kotsu, depart from Kobe Sannomiya station. And if you prefer to leave by train, Osaka Kansai Airport (KIX) to Kagoshima train services, operated by Japan Railways Limited Express, depart from Kansai Airport station.