None of these substitutions are easy. And the result will never be quite like the original. But, happily, they can be better.
First, a WARNING: Common egg substitutes found in supermarkets are not egg-free. They're typically just dried eggs to use when you don't have fresh ones on hand. So, READ THE LABEL.
But, there are plenty of other egg-avoiding folk out there and they've been playing around as well. Here's what I've tried:
Xantham Gum
1/4 tsp Xantham Gum + 1/4 cup water = 1 egg
Mix XG and water. It will form clumps. Try to beat these out of it, but don't get too upset. Leave it to sit for several minutes. Come back, check on it, stir, beat lumps, leave alone. Repeat until lumps are gone and mixture is roughly the look and consistency of egg whites.
So far I've used this for recipes that only need 1 egg, cookies and bars, and they've worked well. The both came out moist and chewey.
Ener-G Egg Replacer
One Egg = 1 1/2 tsp Egg Replacer to 2 tbsp of water.
To replace egg whites use the same amount as replacing the entire egg, so One Egg White=1 1/2 tsp Egg Replacer to 2 tbsp of water.
To replace the egg yolk stir 1-1/2 tsp Egg Replacer into 1 tbsp of water.
So far I've only tried the full egg replacement. 2 eggs equaled 3 tsp of ER and 4 tbsp water. Resulted in a nice, moist cake. 
Interestingly, the Bette Hagman books have quite a few recipes that call for both ER and eggs. It mentions that the ER has good binding properties and helps replace moisture that's left out in a lot of GF flours. I'll have to experiment with those.
