It's an easy process to get your flea-market find looking as good as these examples....
Remove drawer and door knobs, locks, etc. Wash down the surfaces with a suitable cleaner. Repair any dinks or small holes with a wood filler. Sand the surface to remove the original finish.
Use a paint suitable for wood surfaces. Lightly sand between coats for a smooth finish. Two coats is usually enough.
Once paint is fully dry, protect the surface with a varnish or wax. Once dry, replace or re-new handles etc.