The best way to buys bullet train tickets
Although it's possible to purchase bullet train tickets at the automated ticket machines, I prefer to stand in line at the train ticket office (usually located near the shinkansen entrance in stations). Even though this sometimes entails standing in line for a few extra minutes, I prefer being able to designate what kind of tickets I'm buying and also being able to confirm the departure time and platform number in person.
If you're just buying bullet train tickets for a short trip and are not travelling during a busy season / time of day, the walk-up ticket machines are fine for non-reserved seats. However, I prefer to pay a few hundred extra yen (I believe the fee is +500 Yen) for reserved seats for longer trips / trips taken during busy seasons. It's easy to buy reserved seat tickets; many clerks at the JR stations speak some English, but even if they don't, you should be able to buy reserved-seat tickets if you give your start and end destinations, date and preferred time of travel, and can specify smoking or non-smoking train cars.