Domestic Flights
Japan has many interesting places from the shouter most island to the most northern part so it is hard to get everywhere, but thank heavens there are the Domestic flights that can get you where you want to go and usually they will be the fastest way there, and many times more cheaper than shinkansen (bullet train) would be thanks to the low budget airlines.
Where to?
Although flights might be cheaper, somewhat closer destinations may be faster reached by Shinkansen. However, when it comes to distances like Tokyo-Hiroshima or Tokyo-Hokkaido (half the country) or farther, you're probably better off by plane. Also, the many islands are easily and usually only reached by air plane, such as tropical Okinawa
For an up-to-date list of the possible destinations, please check the respective airline websites. I've listed the most common ones below you can only search through sites like SkyScanner or Google flights just as you would any other flight.
And the LCCs operating in Japan: