Although tools exist to remove red-eye in post-production, what's the best way to avoid it while creating the photograph in the field?
To understand how to prevent red-eye you need to understand what causes it.
Red eye is caused by light from a flash that is close to the lens entering the subjects pupils and bouncing off the rear of the eye back into the lens. (The main cause for the red colour is the blood in the back of the retina). Wikipedia has more info.
To prevent red-eye you need to prevent this light bounce back. There are several ways to achieve this.
Pre-flash. By flashing a bright light prior to taking the main photo. This causes the subjects pupils to narrow and reduces the amount of red eye. Many modern compacts have red eye reduction modes that do this by flashing multiple times before the main flash, or providing a continuous bright light in place of a quick flash.
Use an angled flash to direct the light to bounce of a wall or ceiling to prevent the direct bounce back.
Use a flash that is separate from the camera so the light bouncing back does not bounce directly back down the axis of the lens.
If you have a compact camera it is likely your only option is number 1. With an SLR you have more choices.
Using an off camera flash, or bouncing the flash off the ceiling/wall, will help to avoid red-eye.
For a more in-depth answer this is a good article about red-eye control.
If your subjects are prone to red eye, prefiring a flash pulse before taking the shot will make their iris contract, reducing redeye. Many camera's have this as a setting for the flash.
If you're stuck with a point and shoot camera you can try to soften the flash in various ways: a piece of scotch tape over the flash, a tissue held over it, a piece of paper... all of these will allow a varying amounts of flash to pass through but with diffuse light which will prevent red-eye. You'll need to experiment to know exactly which works best for the subject you're shooting.