Lay out a drop cloth in an open area where there is good ventilation. Place your furniture on the drop cloth. Pull out drawers, detach doors by removing hinges and take off knobs or handles. Set all the hardware and screws aside in a container for safe keeping.
Fill holes with a wood filler if you will be changing any hardware. Use a putty knife to push and pack the filler into the holes and allow it to dry. If you're reusing your old hardware or if your new hardware's holes match up with the old ones, you don't need to fill the holes.
Sand every surface and edge that will be painted with a fine-grain sandpaper. Use an electric sander if possible, as sanding by hand can take a lot of elbow grease. Sand the veneer until it's previously smooth, shiny surface is dull and slightly rough. Wipe the surface with a spray cleaner and damp rags.
Apply a primer. Use a spray primer if you are comfortable using spray paint. Alternatively, pour liquid primer into a paint pan and apply it with a roller. Use a brush, if necessary, to prime any crevices or bulky trim. Allow the primer to dry per the manufacturer's suggestions.
Paint the furniture with an acrylic-latex spray paint or apply liquid paint with a roller. Use a brush, if necessary, to fill in any areas that are not smooth. Let the first coat dry for 30 minutes and apply a second coat if necessary.
Spray or brush on a clear varnish or acrylic sealer to protect the surface. Allow it to dry as long as the manufacturer suggests.
Assess where new hardware will go and drill new holes, if necessary. Replace the drawers, cabinets and hardware.