Buy a Set of Countersink Bits
Drilling a pilot hole for the screw and then creating a recess, or countersink, for the screw head is standard practice on cabinets and furniture projects. The pilot hole bit creates a hole that reduces friction to make screw driving easier, and the countersink allows you to set the screw head flush with or below the surface. For straight-shank screws, the less expensive straight-bit design works fine. For tapered-shank wood screws, use a countersink fitted with a tapered-shank bit.
Countersink bits are available with or without stop collars. An adjustable stop collar lets you set the maximum depth of the countersink for more consistent results. Also, you can hide the screw by drilling a deep countersink, called a counterbore, and gluing a plug into the hole. Countersink drill bits are available in sizes to match screw sizes. If you're an avid woodworker, it's worth buying a full set. Otherwise, a No. 7 or No. 8 will cover the most common screw size.