Disassemble the cabinets, removing the hinges and all other hardware. Lay out the doors and drawer fronts on newspapers.
Use your palm sander and rough 60-grit sandpaper to sand down the existing finish, taking as much of it off as you can.
Repeat the sanding process with medium 100-grit sandpaper, getting the rest of the finish off. Follow up with fine 160-grit sandpaper, getting it smooth and clean.
Brush wood stain over the wood surfaces with a paint brush, working in sections of one or two square feet at a time. For each section, brush it on heavily, let it sit for one minute, then wipe up the excess stain with dry rags.
Let the stain set for eight to 10 hours.
Apply polyurethane gloss a smooth, thin coat with a paint brush, brushing it in the direction of the wood grain. Let it dry for six to eight hours.
Sand the gloss lightly with fine 160-grit sandpaper, dulling the shine so the next coat of polyurethane will adhere. Wipe away the dust with dry rags.
Brush on a second layer of polyurethane. Let it dry, lightly hand-sand it with fine 160-grit sandpaper, then brush on a third coat. Let it dry for 24 hours before reassembling the cabinet.