Clean using Tri-Sodium Phosphate or a mild degreaser for getting any grime and dirt off. Repair scratches with wood filler.
Sand any glossy varnish or coating off. You do not need to sand every bit of previous stain or paint off if it is in good condition. Sanding is going to give the best bond, and always sand high traffic surfaces such as table tops, around doorknobs, or dresser tops. Use an orbital sander with 120 grit to make this much easier. Vacuum and wipe all dust off of the panels before priming.
Prime the cabinets with a quality primer. Kilz Premium (not original) is my general go-to primer. For white, use 2 coats of primer. Allow to dry 24-48 hours.
Paint the dresser with at least 2 coats of paint. You can use either a paint sprayer or brushes and rollers. If brushing, brush with the grain and always from one end to the other. If paint starts to dry, don't go back over it.
Guide to Cleaning Furniture for Painting
Knots - use a shellac based primer
Gooey stuff/adhesive - I like Goo Gone for gooey messes, but make sure to clean off with a degreaser or teaspoon afterwards to remove any residue.
Grease stains - clean with a good degreaser like Simple Green to break up what you can.
Permanent Marker/food coloring/other - Clean as best you can with teaspoon and then prime at least two coats.