Kyushu is literally nine provinces in Japanese. Nine provinces made up the island Kyushu. Those provinces were Chikuzen, Chikugo, Hizen, Higo, Buzen, Bungo, Hyuga, Osumi and Satsuma. Suffix 'zen' means front and 'go' means rear. Province near to Kyoto has suffix 'zen' and province which is further has suffix 'go'. Therefore, Chikuzen, Hizen and Buzen are closer to Kyoto than Chikugo, Higo and Bungo.
Currently Kyushu is made up by Fukuoka, Saga, Nagasaki, Kumamoto, Oita, Miyazaki and Kagoshima prefectures. 9 provinces are now 7 prefectures. The most populous prefecture is Fukuoka has two cities designated by government ordinance of Japan which are over half a million inhabitants. Cities designated by government ordinance has wards inside the cities and have much functions than other cities.
These two cities are Fukuoka; the most populous city in Kyushu and 6th in Japan, and Kitakyushu which was the first city to become the designated city out of three metropolitan areas; Tokyo, Osaka and Nagoya. Currently Kyushu has another cities designated by government ordinance, Kumamoto, a prefectural capital of Kumamoto prefecture.
Other relatively big cities are Oita, Nagasaki and Kagoshima. All of them are prefectural capitals of the prefectures which has same name. Probably Nagasaki is the most famous among foreigners. An industrial city with excellent port, Nagasaki was the place where world's second atomic bomb was dropped on 9 August 1945. Ironically the first target was Kokura(current a part of Kitakyushu) but the B-29 bomber flew to Nagasaki due to bad weather in Kokura.
Two Shinkansen are running in Kyushu. Sanyo and Kyushu. Sanyo is connecting Kyushu and Honshu and many trains continue to Tokyo. Kyushu Shinkansen is running through Fukuoka, Saga, Kumamoto and Kagoshima prefectures, west side of the island. A part of Nagasaki Shinkansen will open in autumn 2022. Two prefectures, Oita and Miyazaki are out of Shinkansen range, but limited express trains are available from two Shinkansen stations, Kokura on Sanyo Shinkansen and Kagoshima-Chuo on Kyushu Shinkansen.
If you want to explore Kyushu, you will need at least a week but you can enjoy major hot springs such as Beppu, Ibusuki and Yufuin.