Cover your work area with dropcloths. Plastic tarps, old sheets and tablecloths will work.
Unscrew the drawer pulls and remove them. Set them in a safe place if you intend to reuse them.
Dust off the dresser with clean, lint-free shop cloths.
Fill a bucket with warm water, and add three to four healthy squirts of a degreasing dish detergent. Dip a clean, lint-free shop cloth into the solution and wring it out until it is wet, but not dripping. Give the dresser and drawers a good scrubbing and let them dry thoroughly.
Apply a chemical stripper to the dresser and drawer fronts according to the manufacturer's instructions. Most are painted on with a brush. Let the stripper work, and don't skimp on the time the label says you should wait.
Scrape off the old varnish or paint with a plastic scraper. Apply a second coat of chemical stripper and re-scrape if necessary.
Sand down the entire dresser and the drawer fronts with 80-grit sandpaper. Work with the grain of the wood, not against it. Wipe away the sanding dust with a clean, lint-free shop cloth.
Sand the dresser and drawers again with 120-grit sandpaper, working with the grain. The smoother the wood, the smoother the new finish will be. Wipe away any sanding dust with a clean, lint-free shop cloth.
Dip the corner of a staining sponge into your stain. These look like regular sponges sewn into a cotton cover, and they give you more control and a smoother finish than a brush. Apply the stain to the dresser and drawer fronts in thin, even coats; feather the edges where the stain overlaps, so that you don't get stripes of thicker stain. Use a brush if you are painting, but feather the edges and keep the coat thin and even. Let the stain or paint dry according to the manufacturer's recommendations.
Apply a second coat of stain or paint, if the color does not have the depth you wanted. Sand very lightly between coats with 120- or higher-grit sandpaper if you like, but it is not necessary. Let the second coat dry thoroughly.
Apply a spray-on sealer that is compatible with your stain or paint. This will add a light sheen and provide a little extra protection from wear and spills. Let the sealer dry.
Reattach or replace the hardware, and replace the drawers.