The key to sugar cookies is hidden in the details:
If the recipe says CHILL or refrigerate the dough, it isn't a suggestion - DO IT!
If the recipe does not say chill, DO NOT immediately start rolling out the dough, let it rest and bind - give the liquid a chance to absorb into the dry ingredients. Yes, that's what mixing does....but mixing only gives it the chance to START to absorb - let the dough rest!
Don't overdo the flour on the rolling surface - take a new knee high stocking, rinse it, let it dry thoroughly, fill it with REGULAR (not self rising flour) and use that to lightly dust your rolling surface - if you over flour, your cookies will be dry, crusty, and covered in excess flour and icing won't stick well.
Invest in a rolling pin that has removable thickness rings that allow you to roll out a consistently thick sheet of dough. If your cookies vary in thickness, you will end up with burnt edges on some and doughy centers on other - plus if the thickness isn't the same on all cookies, they will be harder to ice.
Finally USE PARCHMENT PAPER on your cookie sheets, and USE COOLING RACKS!
The final two tips are to give you an easy release from the cookie sheet AND prevent condensation from softening up to bottom of the cookie which can make the cookies break or fall apart when moving them.
For such a simple cookie, sugar cookies can be great, bland, or just plain bad if you don't take the right steps to make them right!