Tokaido Shinkansen is the oldest and busiest Shinkasen line and it is operated between Tokyo, Nagoya, Kyoto and Osaka. Sanyo Shinkansen is operated between Osaka, Okayama, Hiroshima and Hakata (Fukuoka). Both lines are connected each other. Many trains, “Nozomi” that is the fastest train on Tokaido-Sanyo Shinkansen, run from Tokyo to Hiroshima and some of those run to even Hakata.
Most of Hikari run between Tokyo and Shin-Osaka, But some of those run to Okayama and Hiroshima.
Sanyo Shinkansen connected to Kyushu Shinkansen at Hakata. Many trains, Mizuho and Sakura are operated between Shin-Osaka and KUmamoto or Kagoshima-Chuo directly. Please see the post, Kyushu Shinkansen to get details.
“Fujikawa” is operated between Shizuoka (=静岡) and Kofu (=甲府) via Minobu line (=身延線). If you go to Mt. Fuji (=富士山) from west side, this train takes you to the gateway stations, Fuji or Fujinomiya. And also this is a short cut route from Shizuoka to Kofu. If you want to go to Matsumoto, Kobuchizawa or Lake Suwa from Shizuoka, ride on this train and transfer to Limited Express Super Azusa or Limited Express Azusa. This train is fully covered by Japan Rail Pass.
These trains connect Shinjuku with Kofu (甲府) in Yamanashi Pref, Matsumoto (松本) and Minami-Otari (南小谷) in Nagano Pref. These trains go through the mountain area. You can use these trains to access to Lake Kawaguchi area in Mt. Fuji, Akaishi Moutains (a.k.a. Southern Alps), southern part of Nagano pref. and Hida Mountains (a.k.a. Norther Alps). Especially Hida Mountains are very popular to ski and snowboard. Since Nagano held Winter Olympic Games, this area has been getting well known. And also it is very popular area for climbing and hiking in Japan too! Hida Mountains are located in Nagano pref. So you may think about using Nagano Shinkansen. But Nagano Shinkansen takes you to city of Nagano and northern part of Nagano pref. If you access there, you need to access from Matsumoto in south part of Nagano pref.