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.