I think the biggest problem people have when drawing with crayons is that they regress to being five years old again. They hold the crayon like they're coloring again, instead of applying it like a fine art medium. Another problem is that many people want to make it perform like colored pencils. While similar in application, they are quite different. You should not use crayons and expect them to behave like colored pencils. You'll be setting yourself up for disappointment.
Use good quality crayons (Crayola works well) for the best results. Making the fill color darker around the edges is a good technique for adding dimension. As you work, don't press too hard. It's better to add layers of color than to have too much color that you can't remove. Take care to color evenly and without causing color streaks. When you're finished, place a piece of scrap fabric over the coloring and iron the paper on a high setting. This will heat set the coloring and remove excess wax.