Remove all the shelves, drawers and hardware from the cabinet, using your screwdriver.
Sand and de-gloss the parts of the cabinet that will have new laminate on it (generally, the front-facing parts of the frame), using a vibrating sander and medium sandpaper. Wipe up the dust with a rag.
Measure each section that you're going to laminate with a tape measure, and transfer the dimensions to pieces of veneer, using a pencil. Add about an inch all around to each section. Cut out the pieces with a utility knife, running the knife along the back, through the paper that's covering the adhesive, and then snapping the piece. Leave the paper on it.
Hold the first piece of laminate veneer in front of the area of the where it's going, with an inch overhanging all around. Peel back a little of the paper at the end and press the laminate to the surface. Peel back the rest of the paper, pressing the rest of the laminate to the surface.
Trim off the overhanging laminate with your razor knife at the edges of the frame.
Wrap a piece of wood in a dish towel, and rub it firmly across over the surface of the laminate to secure it.
Repeat for each piece. Reassemble the cabinet.