I've not been, but everything I've been taught or read or heard about about Japan says that public transport is the way to go.
The train is the transport mode of choice on Kyushu. The Kyushu Shinkansen zips across from Fukuoka (Hakata) via Kumamoto to Kagoshima in 1:20, and a spur line to Nagasaki is under construction. Ordinary train lines are slower, but can be interesting in other ways: the JR Hisatsu Line (肥薩線) from Kumamoto via Yatsushiro to Hayato (near Kagoshima) is considered one of the most scenic in Japan, with sightseeing trains and, on Fridays, weekends and holidays, even a steam locomotive running part of the way.
The Kyushu Rail Pass, available only to visitors on tourist visas, offers unlimited travel on JR Kyushu's lines, including the Kyushu Shinkansen but not the San'yo Shinkansen to Hakata. As of 2009, the pass costs ¥16,000 for five days, or ¥7,000 for 3-day version limited to northern Kyushu only. you'll have to travel quite a lot to make this pay off and most visitors, especially those not flying in directly to Kyushu, will find the ordinary Japan Rail Pass a better deal.
Buses serve those parts of Kyushu outside the railway network, but schedules tend to be very limited. There is also a highway bus system paralleling the train network, which you can use through the Raku Bus 3 website. They also offer SUNQ 3-4 day unlimited travel passes: 4-day All Kyushu Pass ¥14,000; 3-day All Kyushu Pass ¥10,000; 3-day Northern Kyushu Pass ¥8,000.
As for the cities:
- Fukuoka: Fukuoka is served by 3 subway lines. The Hakata subway station, located under the JR Hakata Station, can take passengers straight to Fukuoka International Airport (6 min, ¥250), as well as to Tenjin, the city's de facto downtown district, and other major stops. An all day subway pass Ichinichi johshaken costs ¥600, while a ticket to the next station Otonari kippu ¥100;, and most commonly travelled distances ¥200 and up. There are ¥1,000, ¥3,000 and ¥5.000 F Cards (with ¥1,100, ¥;3,300, ¥5,700 value). There are also ¥3,000, ¥5,000 Yokanet cards (with ¥1,100, ¥3,300 value) which can be used on all Nishitetsu services and the subway. And ¥1000, ¥3000, ¥5000 WaiWai cards (with same value to cost but ¥20 discounted for each ride only subway line) which can be used on JR-Kyushu line around Fukuoka-city and the subway. Fukuoka is well served by Nishitetsu buses. Buses around the Tenjin and Hakata area cost ¥100. Outside that area, prices go up slightly to about ¥440 for greater distances.
- Nagasaki: Trams connect most of Nagasaki; they run about every ten to fifteen minutes during the day. The most frequently used lines will be the red (3) and blue (1); the blue and red lines run on the same track from the northern end of Nagasaki as far as the Nagasaki train station, where they split. The blue line continues to the You-me Plaza shopping mall, and later the downtown shopping arcade. A one-way trip is ¥120 and you can get a transfer ticket (乗り継ぎ券 ”noritsugi ken") to continue your trip, if it requires two streetcars. These tickets can only be acquired if you get off at the Tsuki Machi stop. You can save money if you're doing a lot of travel by purchasing a daily pass for the streetcars (¥500) which you can purchase at most major hotels.
- Beppu: The town itself is walkable. The Hells may be a bit far for most to find it convenient. Though, walking is a possibility for those who don't mind it, and are not in a hurry. Buses are also available.