Red eye is caused by flash reflecting off the blood vessels in the eyes. The closer the flash is to the lens, the more red eye you'll get. You don't say what kind of camera you got, but the fix will depend upon the camera and the amount of effort you want to expend. The less flash you have to use, the less red eye, so try, whenever possible to use slower shutter speed and increase ambient lighting. If your flash swivels, you can also use bounce flash off the ceiling, so the flash isn't going directly into the eyes.
Oh, and take photographs of pretty girls on horses whenever possible.
Oh, and take photographs of pretty girls on horses whenever possible.Now that's what I call a very helpful suggestion.
I never thought the idea of being a horse would be so appealing ;)