I was brought up to believe that vacuuming (or hoovering as we say back home) should be done first, as the dust rises when you hoover.
After I got married, my wife was horrified to see me hoovering first. She says that she was told that dusting should be first, as when you wipe down surfaces, some dust will inevitably fall on the floor.
So, my question is this, In what order should the cleaning be done to ensure maximum cleanliness?
Dust first, then vacuum. Though the amount of dust that falls seems rather insignificant to the amount you vacuum up.
Second option: call a housecleaning service and open a beer. 
Scientifically, then. I like the question. I am a scientist and spent 17 years working in the fields of cleanrooms, contamination control, and filtration. The question isn’t trivial and depends on all sorts of interesting things.
I think dust first and vacuum second. But you should be using a thoughtfully designed vacuum cleaner with excellent filters such as a HEPA filter, or, better, a central vacuum system that only exhausts outdoors. Home vacuum cleaners today are often so flawed that they do more harm in certain ways than good. The transparent chamber that keeps stirring the collected dirt is an abomination - you want to collect the dirt quietly and keep it still, not agitate it, which keeps breaking it up and rereleasing it to your environment. You also want lots of filter area, as filtering a given quantity of air flow works much better as you increase the filter area (much more than proportionally to the area, in terms of capture efficiency and ability to handle load and work to overcome airflow resistance). Also, vacuum cleaner exhausts may be concentrated and direct the flow into a stream that aerosolizes dirt elsewhere in the room. I have a shop vacuum on wheels that has large paper bags, a secondary cylindrical cartridge HEPA filter, and an exhaust that I diffused with an old sock so that you can’t feel any air flow more than a foot from the machine itself. I should say that HEPA filters, which are 99.97% efficient at trapping particles of 0.3 um in diameter and more efficient than that for all particles that are either larger or smaller, are really overkill; however, there are so few standards for filter efficiency that are marketed to the general public, that the HEPA standard is typically the only choice with any real meaning; moreover the overkill does little harm.
Also, I think the dusting should be done with something that holds dirt well. A vacuum cleaner dust brush attachment seems pretty good to me. It works pretty thoroughly, and also the large flow of a vacuum cleaner means that there is a sort of a sphere of influence around the brush that draws errant motes back into the cleaner. It is important to give a bit of a scrubbing motion and not to travel very far very quickly, so that sphere of influence can work. I also think the “Swiffer” and similar dusters work well. I think they must be electrets, likely made by hot stretching a strongly isotactic polymer film such as polypropylene in an electric field and cooling it before leaving the field. This freezes a statistically uneven distribution of orientation angles for the methyl groups along one side of the polymer backbone into the sheet. Then they shred this sheet and gather the fiber into a felt. As a result there are zillions of strong but uneven electric fields in the air spaces of this felt. Charged dirt particles are attracted by Coulombic force, and uncharged dirt particles tend to become dipoles subject to electrophoresis due to the nonuniform gradients of the tangled little fields. However, though I am impressed by watching these products work, I never studied them professionally (they came along too late) and have to say this is informed speculation.
I never liked the idea of dusting with a dry cloth or feather duster that just moves dust around. I use a damp cloth and turn it regularly so all the dust gets picked up. Than I vacuum when ever I feel the urge. With dirt, the longer you delay, you can always catch up.
Even though it keeps the dust spinning around, the Dyson design passes my “smell test” - it’s the only vac I ever had that didn’t stink. You know that old dusty vacuum cleaner smell. I’ve heard tales of people who’ve used Dyson vacs as air cleaners - they’d just park it in a smoky room, turn it on, and let it run and in reasonably short order, the air is cleared and doesn’t smell of smoke.
We’ve now got a bagged Hoover upright that claims some sort of HEPA ability, and it stinks, so there’s more than just clean air leaving, despite whatever filtering it does.
Impressive respose Napier, but I’m skeptical about one part.
More efficient for smaller particles also? How does that work? Is 0.3 um optimally difficult to capture?
Most interested of you, ZenBeam. Search the web for the phrase “most penetrating particle size” and you find a wealth of info, but the short version is that particles that are very small are easy to capture because they diffuse so much, like a gas. Their path is very long and complicated as they jump vigorously around due to collisions with individual air molecules. In fact, gasses are easily captured by things like sorbers or condensers for this same reason; the visible fins on an air conditioner evaporator coil are fairly good at collecting water molecules.
The most penetrating particle size depends mostly on the air velocity and the diameter of the fibers of the filter. Higher air velocities favor smaller most penetrating sizes by reducing the effectiveness of diffusion (the particles spend less time inside the filter, and also certain inertial effects are enhanced, at least for the largest particles). Larger fiber diameters favor larger most penetrating sizes by providing larger targets for diffusion (which works best between populations of things with very different sizes) and by reducing the effect of interception, which is a geometrical effect (particle boundaries do not deform like streamlines do as air flows around fibers and the particle and fiber peripheries intercept one another).
For HEPA filters as they are usually applied, 0.3 um is the MPPS. For other filters it can be different.
Thanks for the replies guys.
Shop vac can take care of dusting and vacuuming at the same time. Of course, you might lose some smaller bric-a-brac that way, but oh well
Agree.
I used to own a cleaning service. It was always clean from top to bottom, and dust with liquid on a lint-free towel. Of course we also had vacuums that you could use on some higher up places, like blinds if needed.
Thanks, Napier. I had never come across the concept of “most penetrating particle size” before.
It seems from the answers above that, if you do not have proper, industrial strength cleaning apparatuses, then it doesn’t really matter which way you do it.
Would that be correct?
No, still clean from top to bottom. One of the key notions, spray some water with a little vinegar or window cleaner type solution onto a cloth before you dust. Thence the dust will stick to the cloth and not just float up in the air. Finally vacuum. It’s not the vacuum so much as the filter (although not all vacuums are created equally).