To comb your beard you'll need to start off with some good quality combs. There is a difference between a cheap comb and an expensive comb. For the difference of price and the amount you'll comb your beard, it's a well worth piece of investment.
Basically, you'll want to use the combs to separate the hairs and give them room to breathe. Going through the various layers of your beard comb in an upward motion. After separating the beard hairs, you will then brush the hair back down.
I'd advise you to not leave your beard in the "fluffed up" position as it's not necessarily a natural look. You can use a boar's hair brush, your hands, or a comb to push the beard hair down.
At Beardbrand, we sell some of the best combs on the market made out of cellulose acetate which is a plastic made from planets. They start off as blanks and then are hand cut and don't leave the same jagged edges that an extruded plastic has.
Other guys also swear by wooden combs, ox horns combs, or even metal combs.