Cities (Click each for more details)
Really depends on how much time you have and how fast paced you want to move around. Generally I would recommend this for each city:
Tokyo: 3-6 days
Osaka: 2-3 days
Kyoto: 2-3 days
Day Trips from Osaka/Kyoto: As much as you want (maybe 3 days)
Kanazawa/Takayama Region: 4 days
Sendai/Tohoku: 5 days
Kyushu: At least a week
Hakone: 2 days
Izu Peninsula/Shizuoka: At least 5 days
If you want to do the standard trip for your first time, something like a week in the Kansai area (Osaka/Kyoto) and then a week in Tokyo is pretty reasonable. Can shorten each area by a few days to fit in a few days in another region if you want.
Trains.
There is the Shinkansen (high speed bullet train) between most major cities. There are also Limited Express trains between less popular destinations. You don’t need to prebook for this and can generally just head to the station 30ish mins before you want to leave and you can buy a ticket.