I love how you just casually tossed in the “after climbing the Moqui Dugway” part. I’ve been in a car where one person laid down and moaned the entire way up. It’s an 1100 foot vertical climb in three miles, and those three miles are back and forth on the face of a cliff.
If you get a chance I would recommend Little Wild Horse Canyon and Bell Canyon. They make a nice 8-mile round trip non-technical hike and a truly unique hiking experience. There are several dozen other good hike-able slot canyons, but this is a favorite. And the nearby Goblin Valley State Park is an interesting place to poke around as well. And may make a nice day between Moab and Capital Reef, if you end up driving that way.
Wherever you end up, you should definitely find a way to fit at least one slot canyon hike into the mix.
Definitely take a UV umbrella. You may feel silly but walking around in the shade will get you envious looks, believe me. Virtually all of the trails are easily navigable and don’t require climbing. I’ve been to Arches and Canyonlands many times and I don’t think I ever had to use my hands to climb a trail. Strangely enough, I’ve only seen foriegners with the umbrellas, Americans seem to shun them.
OK more safety talk! Going in September is one of the hottest months to be out there in the desert. It’s hot. Hot and dry. Like the inside of a car on a summer day, only it’s all the time. It’s all sandstone and it’s exactly like walking on the world’s largest pizza stone. STAY ON THE PATHS! All that beautiful scenery can look the same, and getting lost can easily be a death sentence. It’s also easy to misjudge when you may be walking close to a cliff. Also watch out for rattlesnakes. As long as you walk normally they will get out of your way, and don’t be dumb and poke into holes or lift up rocks or anything. They get mad when you do that.
Water’s been covered so I won’t mention it again. Even for day hikes take at least a full day’s worth of trail mix or Clif bars and a first aid kit. A foil emergency blanket takes up little space and if you got lost it’d help keep you warm at night.
No one’s suggested doing the Delicate Arch hike yet?? It’s a must see while in Arches National Park. Also Double Arch and Landscape Arch are the touristy things to see, I recommend them. Also take a gander at Brigham’s Unit 
Maps for the various parks will have information on all the major trails. In Moab, definitely check out the Moab brewery for dinner.
I’ll second Delicate Arch. I’ve been there a couple of times, and it’s gorgeous. Also take a look at Wolfe Ranch there, and imagine living in that house. One of the kids who grew up there visited after the place got turned into a national park, which must have been weird.
You’ll get a strange vibe seeing Arches, because they filmed he opening of the film Indiana Jones and the Temple of Doom there. In fact, the very first shots are of Courthouse Towers, which is right off the park road. Notice how the sequence of rocks they pass in subsequent shots are nowhere near each other, but you’ll recognize them.
While in Moab, you might want to see Hole in the Rock:
Former Utahoo checking in. To reinforce: take lots of water and a way to carry the water. Be sure to have some salty snacks on hand. Get a good hat. Pay attention to the weather forecasts and study up on flash floods. A little rain that day doesn’t mean you can’t hike somewhere, but be aware and choose your hikes accordingly. Start getting those boots broken in.
When my wife would visit from New England she always had a bit of trouble with the altitude. You will feel it. I would strongly recommend at least two days for Bryce Canyon. Zion you can easily do in a day. Unless your driver has an extreme fear of heights, enter Zion from the East Entrance. Very memorable. As long as you’re going to drive through Monument Valley, I’d say hit Mesa Verde too.
I notice you haven’t mentioned going out to the North Rim of the Grand Canyon. It’s a very pretty drive. I would recommend spending a day on this.
Are you flying into SLC or elsewhere? Renting a car? FWIW, you’ll probably want to plan your trip stays in groups, X days in Moab, X days near Zion and Bryce, etc. Also, I would definitely recommend getting a National Park pass.
It will save you money in the long run and the you can go to Adams National Park when you get back. 
Sorry, this isn’t the best organized post. I’d be happy to answer any questions you may have.
Imagining all the people down the centuries sniggering at that and observing what it reminds them of is making me smile 
Thanks for the Delicate Arch rec, EvilTOJ and CalMeacham - we’re definitely going to see it.
Well, the airport we fly into is still on the planning board but current thinking is to fly into Denver and then drive the rest of the way. The connections from the UK mean we can fly direct to Denver but SLC needs a connecting flight. Ditto we’re still arguing (well, discussing) the Grand Canyon. I’m kind of thinking of skipping it in order to concentrate on the other parks. Don’t get me wrong - I think it would be mindblowing but I (maybe unfairly?) think it looks SO enormous that I wonder if the sheer scale of the thing makes it harder to appreciate as such. We’ll see!
Whoah! That’s an impressive looking road! Still, as long as it’s passable for cars to pass the other direction without requiring me to risk plummeting those 1100 feet in seconds, I can deal, I think. Thanks for the heads up, though - the ‘how much worse will this road get before it gets better?’ feeling can be most disconcerting in unfamiliar territory! Thanks for the Muley Point rec, ThisOneGuy - the feeling of having an amazing view all to yourself is a great thing. We;ll check it out. Thanks too for the Hike recs, Bartman - I’ll be sure to look into those too.