In general, always work up in aggressiveness of the cleaner, making sure you don't cause more harm than good. Dish soap is a good first level, followed by a Citrus Cleaner (I like Citrus-Solv). A scrub brush can then be added - nylon first and then a brass wire brush if the surface can handle it. If that doesn't work, you can try some Bar Keepers Friend which is mildly abrasive and gets rid of some stains (rust). You can then try solvents like alcohol and acetone, the latter will remove most finishes. If mild cleaning and solvents don't work, you probably then need to remove a layer from the surface with the appropriate abrasive: brass wool, 3M pads, wet sandpaper, rubbing/polishing compound etc.
Wood can be restored pretty easily by cleaning, sanding down to good wood (careful with veneers) and refinishing with oil or varnish. For the painted surfaces, your best bet would be to clean them up, scrape any loose paint off, feather out the chips as best possible and repaint. You can use a good quality Latex Bathroom paint in the interior - easy and cheap.