Now it's easier than ever to buy shinkansen tickets. There are automated ticket machines, in English!, that will guide you to buying your tickets for the shinkansen.
How do you do this? Just go to any station that has a みどりの窓口, midori no mado guchi (green window) and look for the ticket machine. Press the English button, and follow the prompts. You can buy non-reserved, reserved, green car, non-smoking, smoking, window or aisle just by pressing the buttons of your choice. You can even buy tickets in advance, and select the day and time you want to leave. You're also given the choice of number of people, in any combination/number of adults/children. One-way and round-trip tickets are also a choice.
Although I used cash to pay for my tickets, you do have the option of a credit card. I don't know if it will take foreign cards though.
I'd check http://www.hyperdia.com first to see when the time schedule is, and which station(s) actually has a shinkansen stop. I'll be taking the train from Shin-Yokohama to Kyoto. I bought my ticket for June 4th, round-trip, and the ticket is good for 8 days to come back on. I chose the Hikari instead of the Nozomi so I could possibly see some of the countryside.