If you live outside of Japan, you can get a Japan Rail Pass before visiting. This is one of the best deals in Japan. With the JR Pass you can travel on all JR train and JR Bus lines, including the bullet train for a week or if you wish 2 weeks or even 3 weeks, as many times as you want. A weeks costs about $250 and the upper class version (Green Car) costs $350. I recommend the Green Car pass, it gives you more flexibility, the seats roomier, and less crowded. If the trains are full, the Green Car is the last to fill up. Also you occasionally bump into celebrities. On our trip, a couple of times we had the Green Car on the bullet train to ourselves. Although in one case, the Green Car was full – tour group, but the coach cars were empty.
The only restriction is that you cannot use the Nozomi version of the bullet train, which is used between Tokyo and Osaka. However, you can use the Hikari version of the bullet train between Tokyo and Osaka, which run less frequently – once ever 30 minutes as opposed to once every 5 minutes and only a few minutes slower.
A couple of years ago we made use of the overnight trains, Cassiopeia from Tokyo to Hokkaido, and then the Twilight Express from Hokkaido to Osaka. Back then the Twilight Express allowed smoking in the lounge car but now all trains are non-smoking – which will make the trip even more pleasant since the lounge cars are so nice.
JRパスのおかげで、こんなに日本のあちこちにいけるのは間違いない。4万足らず（グリーンの場合で普通なら3万弱）で日本全国どこでも一週間乗り放題。新幹線ものぞみ以外はオッケー。ただ、寝台車は寝台料金は払わないといけないけど、それでも運賃はパスに含まれるので、北斗星も、カシオペアもトワイライトエクスプレスも乗りました。トワイライトエクスプレスは札幌から乗ったので、今度は逆向きをやりたいなあ。だって、そうじゃないとみられないのよ、日本海に沈む夕日。このペンギンとワニが見ているのは函館あたりを過ぎた朝日です。