Miyazaki is a tourism and resort area in southern Kyushu. It is not to be confused with the legendary Japanese animator and filmmaker, Hayao Miyazaki. (Who is from the intellectually-inspired Bunkyo ward of Tokyo, anyway.)
Miyazaki has its own airport. JAL, ANA and the discount carrier Solaseed (formerly Skynet) have regular flights from Tokyo Haneda Airport, Osaka Itami Airport and various other locations throughout Japan. There are also flights to Seoul and Taipei through Asiana and EVA respectively. Airline pricing is highly variable, but the cheapest option to Miyazaki is usually the Solaseed ¥10,000 "Visit Japan" fare for travelers with a “Temporary Visitor” visa (the same requirement as the JR Pass) which is good for just about any single flight on the airline. Though, this is esentially a standby ticket, so you may have trouble during peak demand periods. Peak demand in Japan is around New Years, Golden Week (end of April/start of May), and Obon in August, and of course, Friday and Sunday evenings are always more busy than weekdays.
Buses connect the airport to the main Miyazaki station in 25 minutes (¥400), while the JR Miyazaki Airport Line runs 1-2 trains per hour to Minami-Miyazaki station (¥340). If you are under 21, you can get tickets cheaper than normal.
Most people will take the Kirishima limited express which makes multiple daily runs from Kagoshima (2-2.5 hours, ¥4100).
If you connect to the Shinkansen in Kagoshima, it is anouther 2 hours for Hakata to Kagoshima (2.5 hours from Kukura). You can take a single Shinkansen from Shin-Osaka to Kagoshima in 4.5-5 hours.
Anouther option, the more scenic route, from Kokura station in Kitakyushu, the first stop in Kyushu on the San'yo Shinkansen, is to take the Sonic limited express to Oita or Beppu and transfer to the Nichirin limited express (¥8300). A few Nichirin trains run from Kokura or Hakata in Fukuoka directly to Miyazaki, with no change of trains necessary, but most people that take this route like to stop to see Beppu. The route is about 5 hours which is about the same as the Shinkansen route from Kokura.
Note that there was also an overnight service, the Dream Nichirin, which ran over the same route, however this train service was discontinued on March 11, 2011 (in conjunction with the opening of the completed the Kyushu Shinkansen).
There is no charge for the above services with a Japan Rail Pass, and there is a JR All Kyushu Pass that covers all routes as well, aside from the reference to the Shinkansen in Kokura and beyond. The Kyushu Shinkansen stops at Hakata; Hakata to Shin-Osaka (including Kokura) is owned by JR West, so you would need a JR West Pass for that section.
Starting on March 12, 2011, a highway bus operated by JR Kyushu began to Miyazaki from the Shin-Yatsushiro bullet train station. The "B&S Miyazaki" (B&S for Bus and Shinkansen) will operate once per hour, timed with the arrival of the bullet trains from Fukuoka. The bus trip takes just over 2 hours from Shin-Yatsushiro at a cost of 4,150 yen each way. The overall journey from Fukuoka to Miyazaki will be shortened to just over 3 hours using this train/bus combination. The bus is not valid with the Japan Rail Pass, but bus tickets can be purchased at any JR Midori-no-Madoguchi seat reservation counter.
Hourly buses run from Fukuoka to Miyazaki in about 4 1/2 hours at a cost of ¥6000.
Miyazaki is one of the most famous places for surfing in Japan. Lots of foreigners visit there during the summer. There are good waves and the sunset is so beautiful.
"Kurokirishima" is a famous Miyazaki shochu. Miyazaki produces lots kind of shochu, so people who like Japanese shochu, Miyazaki is a good place to visit. However, some of them are expensive, so you can not buy them so easily.
In the town of Aya, which is around 20km from Miyazaki City proper, there is a complex called the Aya Shusen-no-mori. There is a winery, beer brewery and shochu distillery all on site. The beer made here is German-style, and of a high standard. The Shusen-no-mori brewery makes (at time of writing) a Bock, a Koelsch, an Altbier and a blueberry-flavoured lager. The beer is far superior to the macro-brewed Japanese lagers such as Asahi Super Dry or Kirin Lager. The wine from the Shusen-no-mori winery, on the other hand, leaves something to be desired.
The best hotels in Miyazaki City are the Kanko Hotel, in the city centre by the river, and the Sheraton, in the Seagaia complex surrounded by a golf course around 8km North-East of the city centre. The prices of these luxury hotels are comparable to those of five-star hotel chains in any large Western country.
A cheap place to sleep, on the other hand, is a manga coffee shop/internet cafe. There is one beside the 7-11 near the Miyazaki train station. There is also a "Cybac" internet cafe in Miyazaki which is slightly better than the aforementioned manga kissa - it is difficult to find, so get a taxi from the train station. It should cost around 1000 yen.