How do I avoid red-eye in my pictures?

The best way to avoid red-eye is to have the flash as far away from the lens as possible. If you can mount an external flash onto the flash bracket, use that instead of a built-in or pop-up flash. Similarly, you can use bounce mode to make the light fall on the subject at a different angle/direction from the lens.
If you can get away with raising the ISO, widening the aperture, and/or lowering the shutter speed so you can do without a flash, you can avoid red-eye altogether.