Polish with a Microfiber Cloth
Microfiber cloths are perfect for this because they pick up dust, wipe off smudges and don't shed any fibers. You'll find microfiber cloths wherever cleaning supplies are sold and they'll help you know how to clean your bathroom better. You can also buy them in bulk at wholesale clubs and use them throughout your house for all kinds of other cleaning chores. They're one of the best home cleaning products you can get.
Vacuum First, Then Scrub
For a really thorough cleaning, start at the top, vacuuming the dust from light fixtures and the top of window casings. Then work your way down. And finally, vacuum the floor methodically so you cover every inch. You don't want to leave any stray hair or dust bunnies to muck up your cleaning water. A soft-bristle upholstery brush works best for this type of vacuuming. Figure out what vacuum to use here.
Buff Off Heavy Grime
Pick up some polishing compound at a home center or an auto parts store and use an auto buffer to polish off the offending scum. If you don't own a buffer, you can buy one for as little as $20 or borrow one from a gearhead friend. If possible, remove the doors and take them out to the garage to avoid messing up the bathroom.